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Getting a strong foundation on Calculus

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  • ifeanyi.ogbuokiri@yahoo.com
    Please how would be the easiest way to solve integration of trigonometric ratios Sent from my BlackBerry wireless device from MTN
    Message 1 of 10 , Mar 11, 2012
      Please how would be the easiest way to solve integration of trigonometric ratios

      Sent from my BlackBerry wireless device from MTN
    • Andy Robertshaw
      Hi There!   Integration of trigonemtry ratios is not difficult.   Sin(x) integrates to give - Cos(x)   Cos(x) integrates to give Sin(x)   And Tan(x)
      Message 2 of 10 , Mar 11, 2012
        Hi There!
         
        Integration of trigonemtry ratios is not difficult.
         
        Sin(x) integrates to give - Cos(x)
         
        Cos(x) integrates to give Sin(x)
         
        And Tan(x) integrates to give -ln(Cos(x))
         
         
        Add a constant to all in each case if you are not integrating between limits.
         
        There are other techniques to integrate more complex trig functions, such as integration by parts, by subsitution, partial fractions, and reduction formulae.
         
        Don;t forget as well that Cos(2x) = Cos^2(x) - Sin^2(x) = 1 - 2 Sin^(x) = 2 Cos^2(x) -1. So if you have high powers of trig ratios, then you can convert into lower powers, with larger x-coefficients inside the ratio.
         
        What exactly do you need to solve?
         
        Andy Robertshaw


        ________________________________
        From: "ifeanyi.ogbuokiri@..." <ifeanyi.ogbuokiri@...>
        To: MentalCalculation@yahoogroups.com
        Sent: Sunday, 11 March 2012, 12:07
        Subject: [Mental Calculation] Getting a strong foundation on Calculus



         

        Please how would be the easiest way to solve integration of trigonometric ratios

        Sent from my BlackBerry wireless device from MTN




        [Non-text portions of this message have been removed]
      • Diosdado |Fragata
        Integral calculus is simply the summation of elements. With computers,this mathematical operation could be easily done not just to find the particular solution
        Message 3 of 10 , Mar 11, 2012
          Integral calculus is simply the summation of elements. With computers,this mathematical operation could be easily done not just to find the particular solution to the differential equation but also to provide the solutions at given intervals which is very important in creating tables. The process  discussed below applies to the integration of all differential equations:
           
          MATHEMATICS FOR
          COMPUTERS
                
          The mathematical operations often taught in school deal with the search for a specific solution to a problem. In actual applications however, the progression of values are sometimes more important than the final end result such that the operations must be repeated over and over to yield a table.  Integral Calculus deals with the valuation of the sum of differential elements. It provides the following results:
           
          1.    The particular solution of elements
          2.    The sum of the elements within given limits
           
          Integration can be done mechanically by actual addition of successive elements. Some of these mechanical integration methods are;
           
          a.    SIMPSON,S rules
          b.    TRAPEZOIDAL rule
          c.    TCHEBYCHEFF’S rules
           
          In computers and programmable calculators, the summation of elements can be done successively, allowing the tabulation of values at given intervals to create a table that may be needed for day-to-day usage.
           
          Example:
           
                            Make a table of the volume vs. sounding (depth of contained liquid) of a horizontally mounted cylindrical tank of radius R, and Length L at 1inch interval.
           
           For the cross sectional area of the tank, the equation is:
           
                                                          x2 + y2 = R2    
          Let:  
            Hi= current sounding or level of contained liquid measured from the bottom of the tank
                       Hi-1 = previous sounding or level
                        dh = the thickness of the element of area which is equal to the tabulation interval
                       yi =  current value of y
                       yi-1= previous value of y
                                    xi  = current value of x
                                    dAi= current element of area
                                    dVi = current element of volume
                                    Vi = volume of contained liquid at Hi
                                    Vi-1 = volume at Hi-1
           
          The algebraic relationships of the above variables are:
           
                                  yi = Hi – R
                       
                                  xi  = (R2 – yi2)1/2
           
                                  dAi = (xi + xi-1) dh
           
          dVi = L x dAi
           
          V = Vi-1 + dVi
           
                     
           
          The tabulation may now be done as follows:
           
          H yi xi dVi Vi
          (Hi-1 + dH) (H-R) (R2-yi2)1/2 L(xi – xi-1) dh (Vi-1 + dVi)
          1 (1-R)      
          2        
                 
                   
          2R R      
           
          The above tabulation is fairly accurate although it does not provide a precise value of the volume at a given sounding due to the fact that the value of pis not used. But if the interval is too small compared to the radius of the tank, say 1-inch interval for a radius of 3 feet, the tabulated values would be within the normally accepted values if pis used.
           
          The above operations can be done easily using available computer programs. This method can be used for most engineering calculations where the generation of a table is needed.
           
           BTW, this and other helpful mathematical processes are contained in my books ALTERNATIVE APPROACH TO MATHEMATICS VOL. I and VOL.II which are available for purchase at http//i-proclaimbookstore.com.
           
           

          From: "ifeanyi.ogbuokiri@..." <ifeanyi.ogbuokiri@...>
          To: MentalCalculation@yahoogroups.com
          Sent: Sunday, March 11, 2012 5:07 AM
          Subject: [Mental Calculation] Getting a strong foundation on Calculus


           
          Please how would be the easiest way to solve integration of trigonometric ratios

          Sent from my BlackBerry wireless device from MTN




          [Non-text portions of this message have been removed]
        • AWAP Bouman
          Dear fellow calculators, Reading this all I wonder if we here are still doing mental calculation - in my feelings we are not - or doing higher mathematics.
          Message 4 of 10 , Mar 12, 2012
            Dear fellow calculators,

            Reading this all I wonder if we here are still doing mental calculation - in my feelings we are not - or doing higher mathematics.

            Best regards,



            Willem Bouman


            ----- Original Message -----
            From: Diosdado |Fragata
            To: MentalCalculation@yahoogroups.com
            Sent: Monday, March 12, 2012 2:25 AM
            Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus



            Integral calculus is simply the summation of elements. With computers,this mathematical operation could be easily done not just to find the particular solution to the differential equation but also to provide the solutions at given intervals which is very important in creating tables. The process discussed below applies to the integration of all differential equations:

            MATHEMATICS FOR
            COMPUTERS

            The mathematical operations often taught in school deal with the search for a specific solution to a problem. In actual applications however, the progression of values are sometimes more important than the final end result such that the operations must be repeated over and over to yield a table. Integral Calculus deals with the valuation of the sum of differential elements. It provides the following results:

            1. The particular solution of elements
            2. The sum of the elements within given limits

            Integration can be done mechanically by actual addition of successive elements. Some of these mechanical integration methods are;

            a. SIMPSON,S rules
            b. TRAPEZOIDAL rule
            c. TCHEBYCHEFF’S rules

            In computers and programmable calculators, the summation of elements can be done successively, allowing the tabulation of values at given intervals to create a table that may be needed for day-to-day usage.

            Example:

            Make a table of the volume vs. sounding (depth of contained liquid) of a horizontally mounted cylindrical tank of radius R, and Length L at 1inch interval.

            For the cross sectional area of the tank, the equation is:

            x2 + y2 = R2
            Let:
            Hi= current sounding or level of contained liquid measured from the bottom of the tank
            Hi-1 = previous sounding or level
            dh = the thickness of the element of area which is equal to the tabulation interval
            yi = current value of y
            yi-1= previous value of y
            xi = current value of x
            dAi= current element of area
            dVi = current element of volume
            Vi = volume of contained liquid at Hi
            Vi-1 = volume at Hi-1

            The algebraic relationships of the above variables are:

            yi = Hi – R

            xi = (R2 – yi2)1/2

            dAi = (xi + xi-1) dh

            dVi = L x dAi

            V = Vi-1 + dVi



            The tabulation may now be done as follows:

            H yi xi dVi Vi
            (Hi-1 + dH) (H-R) (R2-yi2)1/2 L(xi – xi-1) dh (Vi-1 + dVi)
            1 (1-R)
            2


            2R R

            The above tabulation is fairly accurate although it does not provide a precise value of the volume at a given sounding due to the fact that the value of pis not used. But if the interval is too small compared to the radius of the tank, say 1-inch interval for a radius of 3 feet, the tabulated values would be within the normally accepted values if pis used.

            The above operations can be done easily using available computer programs. This method can be used for most engineering calculations where the generation of a table is needed.

            BTW, this and other helpful mathematical processes are contained in my books ALTERNATIVE APPROACH TO MATHEMATICS VOL. I and VOL.II which are available for purchase at http//i-proclaimbookstore.com.



            From: "ifeanyi.ogbuokiri@..." <ifeanyi.ogbuokiri@...>
            To: MentalCalculation@yahoogroups.com
            Sent: Sunday, March 11, 2012 5:07 AM
            Subject: [Mental Calculation] Getting a strong foundation on Calculus


            Please how would be the easiest way to solve integration of trigonometric ratios

            Sent from my BlackBerry wireless device from MTN

            [Non-text portions of this message have been removed]





            [Non-text portions of this message have been removed]
          • office
            Hello alltogether, I agree fully with Willem. As I have said before, stuff like this certainly will frighten off beginners, who are looking into the group.
            Message 5 of 10 , Mar 12, 2012
              Hello alltogether,

              I agree fully with Willem. As I have said before, stuff like this certainly will
              frighten off beginners, who are looking into the group. Differential logartihms
              - or whatever it was - has nothing to do with mental calculation in the form
              this group is meant for.

              Best wishes
              werbeka/Bernhard Kauntz




              AWAP Bouman <awap.bouman@...> hat am 12. März 2012 um 15:10 geschrieben:

              > Dear fellow calculators,
              >
              > Reading this all I wonder if we here are still doing mental calculation - in
              > my feelings we are not - or doing higher mathematics.
              >
              > Best regards,
              >
              >
              >
              > Willem Bouman
              >
              >
              > ----- Original Message -----
              > From: Diosdado |Fragata
              > To: MentalCalculation@yahoogroups.com
              > Sent: Monday, March 12, 2012 2:25 AM
              > Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus
              >
              >
              >
              > Integral calculus is simply the summation of elements. With computers,this
              > mathematical operation could be easily done not just to find the particular
              > solution to the differential equation but also to provide the solutions at
              > given intervals which is very important in creating tables. The process
              > discussed below applies to the integration of all differential equations:
              >
              > MATHEMATICS FOR
              > COMPUTERS
              >
              > The mathematical operations often taught in school deal with the search for
              > a specific solution to a problem. In actual applications however, the
              > progression of values are sometimes more important than the final end result
              > such that the operations must be repeated over and over to yield a table.
              > Integral Calculus deals with the valuation of the sum of differential
              > elements. It provides the following results:
              >
              > 1. The particular solution of elements
              > 2. The sum of the elements within given limits
              >
              > Integration can be done mechanically by actual addition of successive
              > elements. Some of these mechanical integration methods are;
              >
              > a. SIMPSON,S rules
              > b. TRAPEZOIDAL rule
              > c. TCHEBYCHEFF’S rules
              >
              > In computers and programmable calculators, the summation of elements can be
              > done successively, allowing the tabulation of values at given intervals to
              > create a table that may be needed for day-to-day usage.
              >
              > Example:
              >
              > Make a table of the volume vs. sounding (depth of
              > contained liquid) of a horizontally mounted cylindrical tank of radius R, and
              > Length L at 1inch interval.
              >
              > For the cross sectional area of the tank, the equation is:
              >
              > x2 + y2 = R2
              > Let:
              > Hi= current sounding or level of contained liquid measured from the bottom
              > of the tank
              > Hi-1 = previous sounding or level
              > dh = the thickness of the element of area which is equal to
              > the tabulation interval
              > yi = current value of y
              > yi-1= previous value of y
              > xi = current value of x
              > dAi= current element of area
              > dVi = current element of volume
              > Vi = volume of contained liquid at Hi
              > Vi-1 = volume at Hi-1
              >
              > The algebraic relationships of the above variables are:
              >
              > yi = Hi – R
              >
              > xi = (R2 – yi2)1/2
              >
              > dAi = (xi + xi-1) dh
              >
              > dVi = L x dAi
              >
              > V = Vi-1 + dVi
              >
              >
              >
              > The tabulation may now be done as follows:
              >
              > H yi xi dVi Vi
              > (Hi-1 + dH) (H-R) (R2-yi2)1/2 L(xi – xi-1) dh (Vi-1 + dVi)
              > 1 (1-R)
              > 2
              >
              >
              > 2R R
              >
              > The above tabulation is fairly accurate although it does not provide a
              > precise value of the volume at a given sounding due to the fact that the value
              > of pis not used. But if the interval is too small compared to the radius of
              > the tank, say 1-inch interval for a radius of 3 feet, the tabulated values
              > would be within the normally accepted values if pis used.
              >
              > The above operations can be done easily using available computer programs.
              > This method can be used for most engineering calculations where the generation
              > of a table is needed.
              >
              > BTW, this and other helpful mathematical processes are contained in my
              > books ALTERNATIVE APPROACH TO MATHEMATICS VOL. I and VOL.II which are
              > available for purchase at http//i-proclaimbookstore.com.
              >
              >
              >
              > From: "ifeanyi.ogbuokiri@..." <ifeanyi.ogbuokiri@...>
              > To: MentalCalculation@yahoogroups.com
              > Sent: Sunday, March 11, 2012 5:07 AM
              > Subject: [Mental Calculation] Getting a strong foundation on Calculus
              >
              >
              > Please how would be the easiest way to solve integration of trigonometric
              > ratios
              >
              > Sent from my BlackBerry wireless device from MTN
              >
              > [Non-text portions of this message have been removed]
              >
              >
              >
              >
              >
              > [Non-text portions of this message have been removed]
              >

              [Non-text portions of this message have been removed]
            • Diosdado |Fragata
              I m sorry if my post did not constitute a mental calculation process. However I was just trying to provide an answer to the request of the original poster for
              Message 6 of 10 , Mar 12, 2012
                I'm sorry if my post did not constitute a mental calculation process. However I was just trying to provide an answer to the request of the original poster for an integration process.
                 
                The books that I mentioned however provide numerous procedures which are useful in sharpening someone's ability to do mental calculations.
                 
                Regards,


                ________________________________
                From: office <office@...>
                To: MentalCalculation@yahoogroups.com
                Sent: Monday, March 12, 2012 10:34 AM
                Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus


                 
                Hello alltogether,

                I agree fully with Willem. As I have said before, stuff like this certainly will
                frighten off beginners, who are looking into the group. Differential logartihms
                - or whatever it was - has nothing to do with mental calculation in the form
                this group is meant for.

                Best wishes
                werbeka/Bernhard Kauntz

                AWAP Bouman <awap.bouman@...> hat am 12. März 2012 um 15:10 geschrieben:

                > Dear fellow calculators,
                >
                > Reading this all I wonder if we here are still doing mental calculation - in
                > my feelings we are not - or doing higher mathematics.
                >
                > Best regards,
                >
                >
                >
                > Willem Bouman
                >
                >
                > ----- Original Message -----
                > From: Diosdado |Fragata
                > To: MentalCalculation@yahoogroups.com
                > Sent: Monday, March 12, 2012 2:25 AM
                > Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus
                >
                >
                >
                > Integral calculus is simply the summation of elements. With computers,this
                > mathematical operation could be easily done not just to find the particular
                > solution to the differential equation but also to provide the solutions at
                > given intervals which is very important in creating tables. The process
                > discussed below applies to the integration of all differential equations:
                >
                > MATHEMATICS FOR
                > COMPUTERS
                >
                > The mathematical operations often taught in school deal with the search for
                > a specific solution to a problem. In actual applications however, the
                > progression of values are sometimes more important than the final end result
                > such that the operations must be repeated over and over to yield a table.
                > Integral Calculus deals with the valuation of the sum of differential
                > elements. It provides the following results:
                >
                > 1. The particular solution of elements
                > 2. The sum of the elements within given limits
                >
                > Integration can be done mechanically by actual addition of successive
                > elements. Some of these mechanical integration methods are;
                >
                > a. SIMPSON,S rules
                > b. TRAPEZOIDAL rule
                > c. TCHEBYCHEFF’S rules
                >
                > In computers and programmable calculators, the summation of elements can be
                > done successively, allowing the tabulation of values at given intervals to
                > create a table that may be needed for day-to-day usage.
                >
                > Example:
                >
                > Make a table of the volume vs. sounding (depth of
                > contained liquid) of a horizontally mounted cylindrical tank of radius R, and
                > Length L at 1inch interval.
                >
                > For the cross sectional area of the tank, the equation is:
                >
                > x2 + y2 = R2
                > Let:
                > Hi= current sounding or level of contained liquid measured from the bottom
                > of the tank
                > Hi-1 = previous sounding or level
                > dh = the thickness of the element of area which is equal to
                > the tabulation interval
                > yi = current value of y
                > yi-1= previous value of y
                > xi = current value of x
                > dAi= current element of area
                > dVi = current element of volume
                > Vi = volume of contained liquid at Hi
                > Vi-1 = volume at Hi-1
                >
                > The algebraic relationships of the above variables are:
                >
                > yi = Hi – R
                >
                > xi = (R2 – yi2)1/2
                >
                > dAi = (xi + xi-1) dh
                >
                > dVi = L x dAi
                >
                > V = Vi-1 + dVi
                >
                >
                >
                > The tabulation may now be done as follows:
                >
                > H yi xi dVi Vi
                > (Hi-1 + dH) (H-R) (R2-yi2)1/2 L(xi – xi-1) dh (Vi-1 + dVi)
                > 1 (1-R)
                > 2
                >
                >
                > 2R R
                >
                > The above tabulation is fairly accurate although it does not provide a
                > precise value of the volume at a given sounding due to the fact that the value
                > of pis not used. But if the interval is too small compared to the radius of
                > the tank, say 1-inch interval for a radius of 3 feet, the tabulated values
                > would be within the normally accepted values if pis used.
                >
                > The above operations can be done easily using available computer programs.
                > This method can be used for most engineering calculations where the generation
                > of a table is needed.
                >
                > BTW, this and other helpful mathematical processes are contained in my
                > books ALTERNATIVE APPROACH TO MATHEMATICS VOL. I and VOL.II which are
                > available for purchase at http//i-proclaimbookstore.com.
                >
                >
                >
                > From: "ifeanyi.ogbuokiri@..." <ifeanyi.ogbuokiri@...>
                > To: MentalCalculation@yahoogroups.com
                > Sent: Sunday, March 11, 2012 5:07 AM
                > Subject: [Mental Calculation] Getting a strong foundation on Calculus
                >
                >
                > Please how would be the easiest way to solve integration of trigonometric
                > ratios
                >
                > Sent from my BlackBerry wireless device from MTN
                >
                > [Non-text portions of this message have been removed]
                >
                >
                >
                >
                >
                > [Non-text portions of this message have been removed]
                >

                [Non-text portions of this message have been removed]




                [Non-text portions of this message have been removed]
              • George Lane
                Greetings to all.   I would like to add my voice to this conevrsation if I may. Whilst the original post was indeed slightly beyond the general remit of this
                Message 7 of 10 , Mar 13, 2012
                  Greetings to all.
                   
                  I would like to add my voice to this conevrsation if I may. Whilst the original post was indeed slightly beyond the general remit of this group, this could easily have been pointed out at an earlier stage - such as the first reply thereto.
                   
                  Furthermore, the books mentioned may well provide a certain sharpening of mental calculating skills within individual persons; so does my own book which I do not promote within this group. This group is for mental calculations, not for free advertising. If someone asks for book recommendations, all well & good - but I do not believe we should take free advantage when it is not requested.
                   
                  Best regards to all,
                   
                  George Lane

                  --- On Tue, 13/3/12, Diosdado |Fragata <dadofragata@...> wrote:


                  From: Diosdado |Fragata <dadofragata@...>
                  Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus
                  To: "MentalCalculation@yahoogroups.com" <MentalCalculation@yahoogroups.com>
                  Date: Tuesday, 13 March, 2012, 0:50



                   



                  I'm sorry if my post did not constitute a mental calculation process. However I was just trying to provide an answer to the request of the original poster for an integration process.
                   
                  The books that I mentioned however provide numerous procedures which are useful in sharpening someone's ability to do mental calculations.
                   
                  Regards,

                  ________________________________
                  From: office <office@...>
                  To: MentalCalculation@yahoogroups.com
                  Sent: Monday, March 12, 2012 10:34 AM
                  Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus

                   
                  Hello alltogether,

                  I agree fully with Willem. As I have said before, stuff like this certainly will
                  frighten off beginners, who are looking into the group. Differential logartihms
                  - or whatever it was - has nothing to do with mental calculation in the form
                  this group is meant for.

                  Best wishes
                  werbeka/Bernhard Kauntz

                  AWAP Bouman <awap.bouman@...> hat am 12. März 2012 um 15:10 geschrieben:

                  > Dear fellow calculators,
                  >
                  > Reading this all I wonder if we here are still doing mental calculation - in
                  > my feelings we are not - or doing higher mathematics.
                  >
                  > Best regards,
                  >
                  >
                  >
                  > Willem Bouman
                  >
                  >
                  > ----- Original Message -----
                  > From: Diosdado |Fragata
                  > To: MentalCalculation@yahoogroups.com
                  > Sent: Monday, March 12, 2012 2:25 AM
                  > Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus
                  >
                  >
                  >
                  > Integral calculus is simply the summation of elements. With computers,this
                  > mathematical operation could be easily done not just to find the particular
                  > solution to the differential equation but also to provide the solutions at
                  > given intervals which is very important in creating tables. The process
                  > discussed below applies to the integration of all differential equations:
                  >
                  > MATHEMATICS FOR
                  > COMPUTERS
                  >
                  > The mathematical operations often taught in school deal with the search for
                  > a specific solution to a problem. In actual applications however, the
                  > progression of values are sometimes more important than the final end result
                  > such that the operations must be repeated over and over to yield a table.
                  > Integral Calculus deals with the valuation of the sum of differential
                  > elements. It provides the following results:
                  >
                  > 1. The particular solution of elements
                  > 2. The sum of the elements within given limits
                  >
                  > Integration can be done mechanically by actual addition of successive
                  > elements. Some of these mechanical integration methods are;
                  >
                  > a. SIMPSON,S rules
                  > b. TRAPEZOIDAL rule
                  > c. TCHEBYCHEFF’S rules
                  >
                  > In computers and programmable calculators, the summation of elements can be
                  > done successively, allowing the tabulation of values at given intervals to
                  > create a table that may be needed for day-to-day usage.
                  >
                  > Example:
                  >
                  > Make a table of the volume vs. sounding (depth of
                  > contained liquid) of a horizontally mounted cylindrical tank of radius R, and
                  > Length L at 1inch interval.
                  >
                  > For the cross sectional area of the tank, the equation is:
                  >
                  > x2 + y2 = R2
                  > Let:
                  > Hi= current sounding or level of contained liquid measured from the bottom
                  > of the tank
                  > Hi-1 = previous sounding or level
                  > dh = the thickness of the element of area which is equal to
                  > the tabulation interval
                  > yi = current value of y
                  > yi-1= previous value of y
                  > xi = current value of x
                  > dAi= current element of area
                  > dVi = current element of volume
                  > Vi = volume of contained liquid at Hi
                  > Vi-1 = volume at Hi-1
                  >
                  > The algebraic relationships of the above variables are:
                  >
                  > yi = Hi – R
                  >
                  > xi = (R2 – yi2)1/2
                  >
                  > dAi = (xi + xi-1) dh
                  >
                  > dVi = L x dAi
                  >
                  > V = Vi-1 + dVi
                  >
                  >
                  >
                  > The tabulation may now be done as follows:
                  >
                  > H yi xi dVi Vi
                  > (Hi-1 + dH) (H-R) (R2-yi2)1/2 L(xi – xi-1) dh (Vi-1 + dVi)
                  > 1 (1-R)
                  > 2
                  >
                  >
                  > 2R R
                  >
                  > The above tabulation is fairly accurate although it does not provide a
                  > precise value of the volume at a given sounding due to the fact that the value
                  > of pis not used. But if the interval is too small compared to the radius of
                  > the tank, say 1-inch interval for a radius of 3 feet, the tabulated values
                  > would be within the normally accepted values if pis used.
                  >
                  > The above operations can be done easily using available computer programs.
                  > This method can be used for most engineering calculations where the generation
                  > of a table is needed.
                  >
                  > BTW, this and other helpful mathematical processes are contained in my
                  > books ALTERNATIVE APPROACH TO MATHEMATICS VOL. I and VOL.II which are
                  > available for purchase at http//i-proclaimbookstore.com.
                  >
                  >
                  >
                  > From: "ifeanyi.ogbuokiri@..." <ifeanyi.ogbuokiri@...>
                  > To: MentalCalculation@yahoogroups.com
                  > Sent: Sunday, March 11, 2012 5:07 AM
                  > Subject: [Mental Calculation] Getting a strong foundation on Calculus
                  >
                  >
                  > Please how would be the easiest way to solve integration of trigonometric
                  > ratios
                  >
                  > Sent from my BlackBerry wireless device from MTN
                  >
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                • Diosdado |Fragata
                  Again I m sorry for having caused so much stir with my reply to somebody who apparently needed some help on calculus. I did not mean to promote my books to the
                  Message 8 of 10 , Mar 13, 2012
                    Again I'm sorry for having caused so much stir with my reply to somebody who apparently needed some help on calculus. I did not mean to promote my books to the group but rather to inform the person of something that might be useful to him.
                     
                    In as much as my apologies does not seem to appease the ire of some members, please accept my resignation from membership in this group. 


                    ________________________________
                    From: George Lane <george972453@...>
                    To: MentalCalculation@yahoogroups.com
                    Sent: Tuesday, March 13, 2012 8:44 AM
                    Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus


                     
                    Greetings to all.
                     
                    I would like to add my voice to this conevrsation if I may. Whilst the original post was indeed slightly beyond the general remit of this group, this could easily have been pointed out at an earlier stage - such as the first reply thereto.
                     
                    Furthermore, the books mentioned may well provide a certain sharpening of mental calculating skills within individual persons; so does my own book which I do not promote within this group. This group is for mental calculations, not for free advertising. If someone asks for book recommendations, all well & good - but I do not believe we should take free advantage when it is not requested.
                     
                    Best regards to all,
                     
                    George Lane

                    --- On Tue, 13/3/12, Diosdado |Fragata <dadofragata@...> wrote:

                    From: Diosdado |Fragata <dadofragata@...>
                    Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus
                    To: "MentalCalculation@yahoogroups.com" <MentalCalculation@yahoogroups.com>
                    Date: Tuesday, 13 March, 2012, 0:50

                     

                    I'm sorry if my post did not constitute a mental calculation process. However I was just trying to provide an answer to the request of the original poster for an integration process.
                     
                    The books that I mentioned however provide numerous procedures which are useful in sharpening someone's ability to do mental calculations.
                     
                    Regards,

                    ________________________________
                    From: office <office@...>
                    To: MentalCalculation@yahoogroups.com
                    Sent: Monday, March 12, 2012 10:34 AM
                    Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus

                     
                    Hello alltogether,

                    I agree fully with Willem. As I have said before, stuff like this certainly will
                    frighten off beginners, who are looking into the group. Differential logartihms
                    - or whatever it was - has nothing to do with mental calculation in the form
                    this group is meant for.

                    Best wishes
                    werbeka/Bernhard Kauntz

                    AWAP Bouman <awap.bouman@...> hat am 12. März 2012 um 15:10 geschrieben:

                    > Dear fellow calculators,
                    >
                    > Reading this all I wonder if we here are still doing mental calculation - in
                    > my feelings we are not - or doing higher mathematics.
                    >
                    > Best regards,
                    >
                    >
                    >
                    > Willem Bouman
                    >
                    >
                    > ----- Original Message -----
                    > From: Diosdado |Fragata
                    > To: MentalCalculation@yahoogroups.com
                    > Sent: Monday, March 12, 2012 2:25 AM
                    > Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus
                    >
                    >
                    >
                    > Integral calculus is simply the summation of elements. With computers,this
                    > mathematical operation could be easily done not just to find the particular
                    > solution to the differential equation but also to provide the solutions at
                    > given intervals which is very important in creating tables. The process
                    > discussed below applies to the integration of all differential equations:
                    >
                    > MATHEMATICS FOR
                    > COMPUTERS
                    >
                    > The mathematical operations often taught in school deal with the search for
                    > a specific solution to a problem. In actual applications however, the
                    > progression of values are sometimes more important than the final end result
                    > such that the operations must be repeated over and over to yield a table.
                    > Integral Calculus deals with the valuation of the sum of differential
                    > elements. It provides the following results:
                    >
                    > 1. The particular solution of elements
                    > 2. The sum of the elements within given limits
                    >
                    > Integration can be done mechanically by actual addition of successive
                    > elements. Some of these mechanical integration methods are;
                    >
                    > a. SIMPSON,S rules
                    > b. TRAPEZOIDAL rule
                    > c. TCHEBYCHEFF’S rules
                    >
                    > In computers and programmable calculators, the summation of elements can be
                    > done successively, allowing the tabulation of values at given intervals to
                    > create a table that may be needed for day-to-day usage.
                    >
                    > Example:
                    >
                    > Make a table of the volume vs. sounding (depth of
                    > contained liquid) of a horizontally mounted cylindrical tank of radius R, and
                    > Length L at 1inch interval.
                    >
                    > For the cross sectional area of the tank, the equation is:
                    >
                    > x2 + y2 = R2
                    > Let:
                    > Hi= current sounding or level of contained liquid measured from the bottom
                    > of the tank
                    > Hi-1 = previous sounding or level
                    > dh = the thickness of the element of area which is equal to
                    > the tabulation interval
                    > yi = current value of y
                    > yi-1= previous value of y
                    > xi = current value of x
                    > dAi= current element of area
                    > dVi = current element of volume
                    > Vi = volume of contained liquid at Hi
                    > Vi-1 = volume at Hi-1
                    >
                    > The algebraic relationships of the above variables are:
                    >
                    > yi = Hi – R
                    >
                    > xi = (R2 – yi2)1/2
                    >
                    > dAi = (xi + xi-1) dh
                    >
                    > dVi = L x dAi
                    >
                    > V = Vi-1 + dVi
                    >
                    >
                    >
                    > The tabulation may now be done as follows:
                    >
                    > H yi xi dVi Vi
                    > (Hi-1 + dH) (H-R) (R2-yi2)1/2 L(xi – xi-1) dh (Vi-1 + dVi)
                    > 1 (1-R)
                    > 2
                    >
                    >
                    > 2R R
                    >
                    > The above tabulation is fairly accurate although it does not provide a
                    > precise value of the volume at a given sounding due to the fact that the value
                    > of pis not used. But if the interval is too small compared to the radius of
                    > the tank, say 1-inch interval for a radius of 3 feet, the tabulated values
                    > would be within the normally accepted values if pis used.
                    >
                    > The above operations can be done easily using available computer programs.
                    > This method can be used for most engineering calculations where the generation
                    > of a table is needed.
                    >
                    > BTW, this and other helpful mathematical processes are contained in my
                    > books ALTERNATIVE APPROACH TO MATHEMATICS VOL. I and VOL.II which are
                    > available for purchase at http//i-proclaimbookstore.com.
                    >
                    >
                    >
                    > From: "ifeanyi.ogbuokiri@..." <ifeanyi.ogbuokiri@...>
                    > To: MentalCalculation@yahoogroups.com
                    > Sent: Sunday, March 11, 2012 5:07 AM
                    > Subject: [Mental Calculation] Getting a strong foundation on Calculus
                    >
                    >
                    > Please how would be the easiest way to solve integration of trigonometric
                    > ratios
                    >
                    > Sent from my BlackBerry wireless device from MTN
                    >
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                    >
                    >
                    >
                    >
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                  • ifeanyi.ogbuokiri@yahoo.com
                    Okay, friends what if you are been given something of this nature to integrate. (2x^3 - 4x) ÷ (5x^3 +6) from 2 to 3 Sent from my BlackBerry wireless device
                    Message 9 of 10 , Mar 13, 2012
                      Okay, friends what if you are been given something of this nature to integrate. (2x^3 - 4x) ÷ (5x^3 +6) from 2 to 3

                      Sent from my BlackBerry wireless device from MTN
                    • Chuck Bean
                      How accurate do you need to be?  I would start out by evaluating the function at 2 and 3, and then averaging the results. ________________________________
                      Message 10 of 10 , Mar 17, 2012
                        How accurate do you need to be?  I would start out by evaluating the function at 2 and 3, and then averaging the results.



                        ________________________________
                        From: "ifeanyi.ogbuokiri@..." <ifeanyi.ogbuokiri@...>
                        To: MentalCalculation@yahoogroups.com
                        Sent: Tuesday, March 13, 2012 3:12 PM
                        Subject: Re: [Mental Calculation] Getting a strong foundation on Calculus

                        Okay, friends what if you are been given something of this nature to integrate. (2x^3 - 4x) ÷ (5x^3 +6) from 2 to 3

                        Sent from my BlackBerry wireless device from MTN

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