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Math Help - Area between curves

  1. #1
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    Area between curves

    Find the area between \[f(x)=sin^5(x)\] and \[g(x)=sin^3(x)\] both with domain \[[0,\pi]\].

    On \[[0,\pi]\] the point of intersection is 0 and \[\frac{\pi}{2}\]. So,

    \[\int_{0}^{\frac{\pi}{2}}[sin^3(x)-sin^5(x)]dx

    =\int_{0}^{\frac{\pi}{2}}[(sin^2x)sin(x)-(sin^2x)^2sin(x)]\ dx

    =\int_{0}^{\frac{\pi}{2}}[(1-cos^2x)sin(x)-(1-cos^2x)^2sin(x)]\ dx

    u=cos(x); \ du=-sin(x)

    \int_{0}^{1}[(1-u^2)-(1-u^2)^2]\ du

    = u - \frac{1}{3}u^3 - u + \frac{2}{3}u^3-\frac{1}{5}u^5\biggr|_{0}^{1}

    = 1 - \frac{1}{3} - 1 +\frac{2}{3} - \frac{1}{5} = \frac{2}{15}


    However, the answer in the back of the book says the answer is \frac{4}{15}. Did I do anything wrong?
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  2. #2
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    Yes. You failed to observe that your point of intersection represents also a point of symmetry. When you changed the limits from [0,\pi] to [0,\pi/2], you should have multiplied the entire expression by 2.
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