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Thread: Area between curves

  1. #1
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    Area between curves

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

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

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

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

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

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

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

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

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


    However, the answer in the back of the book says the answer is $\displaystyle \frac{4}{15}$. 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 $\displaystyle [0,\pi]$ to $\displaystyle [0,\pi/2]$, you should have multiplied the entire expression by 2.
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