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Thread: Help me with double integrals

  1. #1
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    Help me with double integrals

    $\displaystyle \int_D \int (2x-y) dA$
    D is bounded by the circle with center the origin and radius 2.
    help me to get started.
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  2. #2
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    Quote Originally Posted by zpwnchen View Post
    help me to get started.
    Well, as that circle's equation is $\displaystyle x^2+y^2=4 \Longrightarrow y=\pm\sqrt{4-x^2}$, we can see that

    $\displaystyle \iint\limits_D (2x-y)dA=\int\limits_{-2}^2\int \limits_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(2x-y)dydx$

    Tonio
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  3. #3
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    Quote Originally Posted by zpwnchen View Post
    help me to get started.
    When dealing with circles, I would suggest converting to polar coordinates.

    Do you know the following formulae:

    $\displaystyle x = r \cos (\theta) $

    $\displaystyle y = r \sin (\theta) $

    Also, do you know the Jacobian matrix for a change of variables in double integration?

    Once you have converted to polar coordinates, your variables of integration will be r, the radius, and $\displaystyle \theta$, the angle. Clearly, the limits for these are as follows:

    $\displaystyle 0 \leq r \leq 2 $

    $\displaystyle 0 \leq \theta \leq 2\pi $
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