1. ## polar coordinates

Hello,
I would really appreciate if someone could check my methodology.Thanks.
∫∫
s xyex^2+y^2dxdy

describe the region in terms of the polar coordinates r and theta.
its is the region defined by x2+y2 less than or equal to 1
x≥0y≥0.

any suggestions ? I did the integral for r first ,where its boundary is from o to 1
theta varies from zero to pi/2

I made a u substituion where u=r^2 clearly du would be ...

dxdxy=rdrdθ

I took x =rcosθ
y=rsinθ

2. ## Re: polar coordinates

Yes, that will work. You will have two integrals, $\displaystyle \int_0^{\pi/2}sin(\theta)cos(\theta)d\theta$, which is easy, and $\displaystyle \int_0^1 r^2 e^{r^2} drdr$ and, as you say, the substitution $\displaystyle u= r^2$ turns that into $\displaystyle \frac{1}{2}\int_0^1 ue^u du$ which can be done with "integration by parts".

3. ## Re: polar coordinates

Originally Posted by HallsofIvy
Yes, that will work. You will have two integrals, $\displaystyle \int_0^{\pi/2}sin(\theta)cos(\theta)d\theta$, which is easy, and $\displaystyle \int_0^1 r^2 e^{r^2} drdr$ and, as you say, the substitution $\displaystyle u= r^2$ turns that into $\displaystyle \frac{1}{2}\int_0^1 ue^u du$ which can be done with "integration by parts".
Actually it would be \displaystyle \displaystyle \begin{align*} \int_0^1{ r^3\,e^{r^2}\,dr} \end{align*}, since you have the extra factor of r when converting from Cartesians to Polars.