Riemann Sum help

[jdom543]

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]

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

[Martingale]

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]

[jdom543]

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

[Martingale]

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