# Math Help - Help with a multiple integral

1. ## Help with a multiple integral

Hello,

I need help integrating this integral. I tried doing it in wolfram alpha but I don't think it should be as complicated as the result it showed was.

$\iint\frac{xy^{2}}{1+y^{4}}dxdy$

Please just explain it to me or help me get started. I'm really not sure what to do.

Thanks!

2. ## Re: Help with a multiple integral

integrate x alone and y alone first begin with x since dx coming first

can you integrate x I think you should know how to do that
then y you have to integrate $\frac{y^2}{1+y^4}$ can you ?

3. ## Re: Help with a multiple integral

Yeah I know how to get the integral with repect to x, but I don't know how to get the integral with respect to y from x^2 to 1

4. ## Re: Help with a multiple integral

I am still trying to solve $\int \frac{y^2 }{y^4+1} \; dy$

but you can see here a wolfram solution it is messy click into show steps

here

5. ## Re: Help with a multiple integral

Originally Posted by tubetess123
Hello,

I need help integrating this integral. I tried doing it in wolfram alpha but I don't think it should be as complicated as the result it showed was.

$\iint\frac{xy^{2}}{1+y^{4}}dxdy$

Please just explain it to me or help me get started. I'm really not sure what to do.

Thanks!
What are your bounds?

6. ## Re: Help with a multiple integral

The bounds are $0 \leq x \leq 1$ and $x^{2}\leq y \leq 1$.

Please show work. I get it down to the integral with just y and have no idea what to do. Might reversing the order of integration work? I really don't know.

Thanks!

7. ## Re: Help with a multiple integral

Originally Posted by tubetess123
The bounds are $0 \leq x \leq 1$ and $x^{2}\leq y \leq 1$.

Please show work. I get it down to the integral with just y and have no idea what to do. Might reversing the order of integration work? I really don't know.

Thanks!
A better idea is for you to show some work. Reversing the order of integration might be a good idea, if the current order is causing problems. By looking at the integrand, it looks much easier to integrate with respect to x first...

What will your new bounds be if you change the order?

8. ## Re: Help with a multiple integral

They should be $0 \leq x \leq \sqrt{y}$ and $0 \leq y \leq 1$.

9. ## Re: Help with a multiple integral

Originally Posted by tubetess123
They should be $0 \leq x \leq \sqrt{y}$ and $0 \leq y \leq 1$.
Correct, so now evaluate

\displaystyle \begin{align*} \int_0^1{\int_0^{\sqrt{y}}{\frac{x\,y^2}{1 + y^4}\,dx}\,dy} \end{align*}

10. ## Re: Help with a multiple integral

But I thought we had to actually switch the order of integration? Doesn't that mean we have to switch dx and dy too? And switch which boundaries are on which integral sign?

11. ## Re: Help with a multiple integral

$\int_0^2 \int_0^{\sqrt{y}} \frac{xy}{1+y^4} \; dx\; dy = \int_{0}^{1} \frac{(\sqrt{y})^2y}{2(1+y^4)} - \frac{0(y)}{2(1+y^4)} \; dy = \int_{0}^{1} \frac{y^3}{2(1+y^4)} dy$
and this can be solved by sub u = 1+y^4

12. ## Re: Help with a multiple integral

Ok I got $\frac{log(2)}{8}$! Does that look right?

13. ## Re: Help with a multiple integral

it is correct
Wolfram

14. ## Re: Help with a multiple integral

Originally Posted by tubetess123
But I thought we had to actually switch the order of integration? Doesn't that mean we have to switch dx and dy too? And switch which boundaries are on which integral sign?
The bounds you gave implied integration with respect to y first, so after you reverse the order, it implies an integration with respect to x first.

15. ## Re: Help with a multiple integral

So whether dx or dy is written first doesn't determine the order of integration?

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