## Riemann Sum help

Limits, differentiation, related rates, integration, trig integrals, etc.
jdom543
Posts: 12
Joined: Tue Dec 01, 2009 3:05 am
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### Riemann Sum help

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

Martingale
Posts: 335
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Riemann Sum help

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

$x_i=a+i\Delta x=0+i\frac{1}{n}=\frac{i}{n}$

Martingale
Posts: 335
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Riemann Sum help

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

$\lim_{n\to\infty} \sum_{i=0}^{n}{f(x_i^*)\Delta x$

Code: Select all

$$\lim_{n\to\infty} \sum_{i=0}^{n}{f(x_i^*)\Delta x$$

jdom543
Posts: 12
Joined: Tue Dec 01, 2009 3:05 am
Contact:

### Re: Riemann Sum help

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

$x_i=a+i\Delta x=0+i\frac{1}{n}=\frac{i}{n}$

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

Martingale
Posts: 335
Joined: Mon Mar 30, 2009 1:30 pm
Location: USA
Contact:

### Re: Riemann Sum help

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

$\int\limits_{a}^{b}f(x)dx=\lim_{n\to\infty} \sum_{i=0}^{n}{f(a+i\Delta x)\Delta x=\lim_{n\to\infty} \sum_{i=0}^{n}{f(a+i\frac{b-a}{n})\frac{b-a}{n}$

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...

$\int\limits_{a}^{b}\left[f(x)-0\right]dx=\int\limits_{a}^{b}f(x)dx$

the top function - the bottom function

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

$\int\limits_{a}^{b}\left[0-f(x)\right]dx=\int\limits_{a}^{b}\left[-f(x)\right]dx$

thus the area tapped between $x^3$ and the x-axis on the interval [-1,0] is given by

$\int\limits_{-1}^{0}\left[0-x^3\right]dx=\int\limits_{-1}^{0}\left[-x^3\right]dx$