# Thread: multiple integrals polar integration

1. ## multiple integrals polar integration

in the question

we have to evaluate this integral by converting into polar integral

i know x=rcosa and y=rsina then we can use jacobian and write dydx as rdrda the only problem is how to find the limits of integration please help

2. ## Re: multiple integrals polar integration

Originally Posted by prasum

Perhaps you haven't quoted it correctly. Take into account that if $\displaystyle 1<x\leq 2$, then $\displaystyle \sqrt{1-x^2}$ is not defined in $\displaystyle \mathbb{R}$.

3. ## Re: multiple integrals polar integration

it is thomas and finney calculus 9th edition question in ex 13.3

4. ## Re: multiple integrals polar integration

Originally Posted by prasum
it is thomas and finney calculus 9th edition question in ex 13.3
Surely it is a typo. I suppose he meant $\displaystyle \int_0^1\int_0^{\sqrt{1-x^2}}\frac{x+y}{x^2+y^2}\;dydx$. In such case, the integration domain is $\displaystyle D\equiv \begin{Bmatrix} 0\leq \theta\leq \pi/2\\0\leq \rho\leq 1\end{matrix}$ .

5. ## Re: multiple integrals polar integration

sorry it is sqrt(1-(x-1)^2) instead of sqrt(1-x^2)

6. ## Re: multiple integrals polar integration

Originally Posted by prasum
sorry it is sqrt(1-(x-1)^2) instead of sqrt(1-x^2)
That is another thing. The integration domain is limited by the circle $\displaystyle (x-1)^2+y^2=1$ (equivalently $\displaystyle x^2+y^2-2x=0$) and $\displaystyle y=0$ (upper semidisk). Using polar coordinates $\displaystyle x=\rho\cos \theta,\;y=\rho\sin\theta$ we get $\displaystyle \rho^2-2\rho\cos\theta=0$ (equivalently $\displaystyle \rho(\rho-2\cos\theta)=0$). As a consequence the integration domain is $\displaystyle D\equiv \begin{Bmatrix} 0\leq \theta\leq \pi/2\\0\leq \rho\leq 2\cos\theta\end{matrix}.$