Mental Calculation is a Public Group with 1188 members.
 Mental Calculation

 Public Group,
 1188 members
Getting a strong foundation on Calculus
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 Please how would be the easiest way to solve integration of trigonometric ratios
Sent from my BlackBerry wireless device from MTN  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 xcoefficients 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
[Nontext portions of this message have been removed]  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 daytoday 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
Hi1 = previous sounding or level
dh = the thickness of the element of area which is equal to the tabulation interval
yi = current value of y
yi1= 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
Vi1 = volume at Hi1
The algebraic relationships of the above variables are:
yi = Hi – R
xi = (R2 – yi2)1/2
dAi = (xi + xi1) dh
dVi = L x dAi
V = Vi1 + dVi
The tabulation may now be done as follows:
H yi xi dVi Vi
(Hi1 + dH) (HR) (R2yi2)1/2 L(xi – xi1) dh (Vi1 + dVi)
1 (1R)
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 1inch 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//iproclaimbookstore.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
[Nontext portions of this message have been removed]  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 daytoday 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
Hi1 = previous sounding or level
dh = the thickness of the element of area which is equal to the tabulation interval
yi = current value of y
yi1= 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
Vi1 = volume at Hi1
The algebraic relationships of the above variables are:
yi = Hi – R
xi = (R2 – yi2)1/2
dAi = (xi + xi1) dh
dVi = L x dAi
V = Vi1 + dVi
The tabulation may now be done as follows:
H yi xi dVi Vi
(Hi1 + dH) (HR) (R2yi2)1/2 L(xi – xi1) dh (Vi1 + dVi)
1 (1R)
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 1inch 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//iproclaimbookstore.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
[Nontext portions of this message have been removed]
[Nontext portions of this message have been removed]  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,
[Nontext portions of this message have been removed]
>
> 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 daytoday 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
> Hi1 = previous sounding or level
> dh = the thickness of the element of area which is equal to
> the tabulation interval
> yi = current value of y
> yi1= 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
> Vi1 = volume at Hi1
>
> The algebraic relationships of the above variables are:
>
> yi = Hi – R
>
> xi = (R2 – yi2)1/2
>
> dAi = (xi + xi1) dh
>
> dVi = L x dAi
>
> V = Vi1 + dVi
>
>
>
> The tabulation may now be done as follows:
>
> H yi xi dVi Vi
> (Hi1 + dH) (HR) (R2yi2)1/2 L(xi – xi1) dh (Vi1 + dVi)
> 1 (1R)
> 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 1inch 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//iproclaimbookstore.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
>
> [Nontext portions of this message have been removed]
>
>
>
>
>
> [Nontext portions of this message have been removed]
>
 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,
[Nontext portions of this message have been removed]
>
> 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 daytoday 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
> Hi1 = previous sounding or level
> dh = the thickness of the element of area which is equal to
> the tabulation interval
> yi = current value of y
> yi1= 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
> Vi1 = volume at Hi1
>
> The algebraic relationships of the above variables are:
>
> yi = Hi – R
>
> xi = (R2 – yi2)1/2
>
> dAi = (xi + xi1) dh
>
> dVi = L x dAi
>
> V = Vi1 + dVi
>
>
>
> The tabulation may now be done as follows:
>
> H yi xi dVi Vi
> (Hi1 + dH) (HR) (R2yi2)1/2 L(xi – xi1) dh (Vi1 + dVi)
> 1 (1R)
> 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 1inch 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//iproclaimbookstore.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
>
> [Nontext portions of this message have been removed]
>
>
>
>
>
> [Nontext portions of this message have been removed]
>
[Nontext portions of this message have been removed]  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 daytoday 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
> Hi1 = previous sounding or level
> dh = the thickness of the element of area which is equal to
> the tabulation interval
> yi = current value of y
> yi1= 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
> Vi1 = volume at Hi1
>
> The algebraic relationships of the above variables are:
>
> yi = Hi – R
>
> xi = (R2 – yi2)1/2
>
> dAi = (xi + xi1) dh
>
> dVi = L x dAi
>
> V = Vi1 + dVi
>
>
>
> The tabulation may now be done as follows:
>
> H yi xi dVi Vi
> (Hi1 + dH) (HR) (R2yi2)1/2 L(xi – xi1) dh (Vi1 + dVi)
> 1 (1R)
> 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 1inch 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//iproclaimbookstore.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
>
> [Nontext portions of this message have been removed]
>
>
>
>
>
> [Nontext portions of this message have been removed]
>
[Nontext portions of this message have been removed]
[Nontext portions of this message have been removed]
[Nontext portions of this message have been removed]  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 daytoday 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
> Hi1 = previous sounding or level
> dh = the thickness of the element of area which is equal to
> the tabulation interval
> yi = current value of y
> yi1= 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
> Vi1 = volume at Hi1
>
> The algebraic relationships of the above variables are:
>
> yi = Hi – R
>
> xi = (R2 – yi2)1/2
>
> dAi = (xi + xi1) dh
>
> dVi = L x dAi
>
> V = Vi1 + dVi
>
>
>
> The tabulation may now be done as follows:
>
> H yi xi dVi Vi
> (Hi1 + dH) (HR) (R2yi2)1/2 L(xi – xi1) dh (Vi1 + dVi)
> 1 (1R)
> 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 1inch 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//iproclaimbookstore.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
>
> [Nontext portions of this message have been removed]
>
>
>
>
>
> [Nontext portions of this message have been removed]
>
[Nontext portions of this message have been removed]
[Nontext portions of this message have been removed]
[Nontext portions of this message have been removed]
[Nontext portions of this message have been removed]  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  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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