1. ## Integration

I was given a problem about revolving a region around the x-axis and finding the volume of the region. I know for sure that this is the correct integral for the particular problem: (notation: exp(x)=e^x)

pi(exp(3x)^2)-pi(exp(x))^2

And the upper bound is 2 and the lower bound is 0.

I split the integral into two separate integrals and I was able to solve the second part (pi(exp(x))^2) but I was told by another math tutor that the first integral couldn't be solved by rules of integration, but instead by estimation. She said simpson's rule would be best, but I thought we needed an "n" to solve for that...and even then I'm not quite sure I understand what she meant. Help!
Thanks a lot!

2. Originally Posted by amaya
I was given a problem about revolving a region around the x-axis and finding the volume of the region. I know for sure that this is the correct integral for the particular problem: (notation: exp(x)=e^x)

pi(exp(3x)^2)-pi(exp(x))^2

And the upper bound is 2 and the lower bound is 0.

I split the integral into two separate integrals and I was able to solve the second part (pi(exp(x))^2) but I was told by another math tutor that the first integral couldn't be solved by rules of integration, but instead by estimation. She said simpson's rule would be best, but I thought we needed an "n" to solve for that...and even then I'm not quite sure I understand what she meant. Help!
Thanks a lot!
$\int e^{x^2}dx$ can't be integrated using elementry functions we could use simpsons rule to approximate it. Remember with approximation rules like simpson we choose n.

We can integrate it by using its power series.

$e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$

so evaluating at $x^2$ we get

$e^{x^2}=\sum_{n=0}^{\infty}\frac{(x^2)^n}{n!}$

integrating we get...

$\int e^{x^2}dx =\int \sum_{n=0}^{\infty}\frac{(x^2)^n}{n!}dx=\sum_{n=0} ^{\infty}\frac{x^{2n+1}}{(2n+1)(n!)}$

3. ## integration..same problem

I was able to solve the (exp(x))^2 because it turns out being Exp(2x) which I can solve. The one I'm having problems with is:

integral from 0 to 2 of: 9(x^2) (nine "x" squared)

help!

4. Someone isn't paying attention. I think it's "another math tutor".

$\int{e^{x^2}}dx$ is a bit of a problem.

$\int{(e^{x})^{2}}dx\;=\;\frac{1}{2}(e^{x})^{2}+C$

$\int{(e^{3x})^{2}}dx\;=\;\frac{1}{6}(e^{3x})^{2}+C$