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Math Help - Evalute the iterated integral..two of them.

  1. #1
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    Evalute the iterated integral..two of them.

    Alright I'm having some trouble with two of the problems in my homework.
    Any help would be awesome.

    1. <br /> <br />
\int_{0}^\frac{\pi}{4}\int_{\sqrt{3}}^{\sqrt{3}\co  s\theta}r dr d\theta<br /> <br />

    2. <br /> <br />
\int_{0}^\frac{\pi}{4}\int_{0}^{\cos\theta}3r^2\si  n\theta drd\theta<br /> <br />
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  2. #2
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    Just in case a picture helps...



    ... where a straight line differentiates downwards (anti-differentiates up) with respect to the letter that labels it.

    Spoiler:

    Spoiler:


    _________________________________________
    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
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  3. #3
    Super Member Random Variable's Avatar
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     \int_{0}^\frac{\pi}{4}\int_{\sqrt{3}}^{\sqrt{3}\co  s\theta}r dr \ d\theta = \frac{1}{2} \int^{\frac{\pi}{4}}_{0} (3 \cos ^{2} \theta -3) \ d\theta = -\frac{3}{2} \int^{\frac{\pi}{4}}_{0} \sin^{2} \theta \ d\theta

     = -\frac{3}{2} \int^{\frac{\pi}{4}}_{0} \Big( \frac{1}{2} - \frac{\cos 2\theta}{2} \Big) \ d \theta = -\frac{3}{2} \Big( \frac{x}{2} - \frac{\sin 2 \theta}{4} \Big) \Big|^{\frac{\pi}{4}}_{0}

     = \frac{-3}{2} \Big(\frac{\pi}{8} - \frac{1}{4}\Big) = \frac{3}{8} - \frac{3 \pi}{16}
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