Riemann Sum help  TOPIC_SOLVED

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Riemann Sum help  TOPIC_SOLVED

Postby jdom543 on Fri Aug 13, 2010 11:56 pm

So I just have a question about Riemann Sums.

I give an example to explain my question.

Let's say I wanted to find the area under a curve of f(x)=x^2 on the closed interval [0,1]. The area would be equivalent to

lim (n-->inifnity) ((n(above)Sigma(i=0) (f(x(subi))*(deltax)))

Hopefully someone can understand that.

It should look something like this

So delta x would be (1-0)/n which is 1/n

But how do I find x(sub i) so I can plug it into f(xsubi)? Is there a universal formula that I may use with Riemann Sums to find x(subi) even for more complex problems?
Thanks
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Re: Riemann Sum help

Postby Martingale on Sat Aug 14, 2010 12:49 am

jdom543 wrote:So I just have a question about Riemann Sums.

I give an example to explain my question.

Let's say I wanted to find the area under a curve of f(x)=x^2 on the closed interval [0,1]. The area would be equivalent to

lim (n-->inifnity) ((n(above)Sigma(i=0) (f(x(subi))*(deltax)))

Hopefully someone can understand that.

It should look something like this

So delta x would be (1-0)/n which is 1/n

But how do I find x(sub i) so I can plug it into f(xsubi)? Is there a universal formula that I may use with Riemann Sums to find x(subi) even for more complex problems?
Thanks


One can use



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Re: Riemann Sum help

Postby Martingale on Sat Aug 14, 2010 12:53 am

jdom543 wrote:lim (n-->inifnity) ((n(above)Sigma(i=0) (f(x(subi))*(deltax)))





Code: Select all
[tex]\lim_{n\to\infty} \sum_{i=0}^{n}{f(x_i^*)\Delta x[/tex]
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Re: Riemann Sum help

Postby jdom543 on Sat Aug 14, 2010 2:18 am

Martingale wrote:
jdom543 wrote:So I just have a question about Riemann Sums.

I give an example to explain my question.

Let's say I wanted to find the area under a curve of f(x)=x^2 on the closed interval [0,1]. The area would be equivalent to

lim (n-->inifnity) ((n(above)Sigma(i=0) (f(x(subi))*(deltax)))

Hopefully someone can understand that.

It should look something like this

So delta x would be (1-0)/n which is 1/n

But how do I find x(sub i) so I can plug it into f(xsubi)? Is there a universal formula that I may use with Riemann Sums to find x(subi) even for more complex problems?
Thanks


One can use




Thanks quick question though. what if the part I wanted to find the area under (or above in this case I guess) was in the -f(x) values. Ex f(x)=x^3 on the closed interval [-1,0] Would 'a' be -1? or 0? in the formula you gave. Also would you have to make i negative since you're dealing with -f(x) values?

Thanks
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Re: Riemann Sum help

Postby Martingale on Sat Aug 14, 2010 2:47 am

jdom543 wrote:Thanks quick question though. what if the part I wanted to find the area under (or above in this case I guess) was in the -f(x) values. Ex f(x)=x^3 on the closed interval [-1,0] Would 'a' be -1? or 0? in the formula you gave. Also would you have to make i negative since you're dealing with -f(x) values?

Thanks





the integral finds the area between the function and the x-axis so when you are finding the area under a positive function (a function above the x-axis) we have...



the top function - the bottom function

if f(x) is negative then the 'top' function is the zero function. so we get




thus the area tapped between and the x-axis on the interval [-1,0] is given by

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