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Math Help - polar coordinates

  1. #1
    n22
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    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
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  2. #2
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    Re: polar coordinates

    Yes, that will work. You will have two integrals, \int_0^{\pi/2}sin(\theta)cos(\theta)d\theta, which is easy, and \int_0^1 r^2 e^{r^2} drdr and, as you say, the substitution u= r^2 turns that into \frac{1}{2}\int_0^1 ue^u du which can be done with "integration by parts".
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  3. #3
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    Re: polar coordinates

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