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Math Help - multiple integrals polar integration

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    multiple integrals polar integration

    in the questionmultiple integrals polar integration-codecogseqn-1-.gif

    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
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    MHF Contributor FernandoRevilla's Avatar
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    Re: multiple integrals polar integration

    Quote Originally Posted by prasum View Post
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    Perhaps you haven't quoted it correctly. Take into account that if 1<x\leq 2, then \sqrt{1-x^2} is not defined in \mathbb{R}.
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    Re: multiple integrals polar integration

    it is thomas and finney calculus 9th edition question in ex 13.3
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    MHF Contributor FernandoRevilla's Avatar
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    Re: multiple integrals polar integration

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

    sorry it is sqrt(1-(x-1)^2) instead of sqrt(1-x^2)
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    MHF Contributor FernandoRevilla's Avatar
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    Re: multiple integrals polar integration

    Quote Originally Posted by prasum View Post
    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 (x-1)^2+y^2=1 (equivalently x^2+y^2-2x=0) and y=0 (upper semidisk). Using polar coordinates x=\rho\cos \theta,\;y=\rho\sin\theta we get \rho^2-2\rho\cos\theta=0 (equivalently \rho(\rho-2\cos\theta)=0). As a consequence the integration domain is D\equiv \begin{Bmatrix} 0\leq \theta\leq \pi/2\\0\leq \rho\leq 2\cos\theta\end{matrix}.
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