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Math Help - Cylindrical Shells Help

  1. #1
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    Cylindrical Shells Help

    R is bounded below by the x-axis and above by the curve y=2cosx,0 \leq x \leq \frac{\pi}{2}. Find the volume of the solid generated by revolving R around the y-axis .by the method of cylindrical shells.
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  2. #2
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    Hello, Yogi!

    R is bounded below by the x-axis and above by the curve y = 2\cos x,\;0 \leq x \leq \frac{\pi}{2}.
    Find the volume of the solid generated by revolving R around the y-axis
    by the method of cylindrical shells.

    The shells formula is: . V \:=\:2\pi\int^b_axy\,dx

    We have: . V \;= \;2\pi\int^{\frac{\pi}{2}}_0x\cdot2\cos x\,dx \;=\;4\pi\int^{\frac{\pi}{2}}_0x\cos x\,dx

    Integrate by parts:
    . . \begin{array}{cc}u = x & dv = \cos x\,dx \\ du = dx & v = \sin x\end{array}

    We have: . V \:=\:4\pi\left[x\sin x - \int\sin x\,dx\right] \:=\:4\pi\bigg[x\sin x + \cos x \bigg]^{\frac{\pi}{2}}_0

    . . = \;4\pi\left[\left(\frac{\pi}{2}\!\cdot\!\sin\frac{\pi}{2} + \cos\frac{\pi}{2}\right) - \left(0\!\cdot\!\sin0 + \cos0\right)\right]

    . . = \;4\pi\left[\left(\frac{\pi}{2} + 0\right) - \left(0 + 1\right)\right] \;=\;4\pi\left(\frac{\pi}{2} - 1\right)\;=\;2\pi(\pi - 2)

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  3. #3
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    We can also check by doing it the 'other' way....washers.

    {\pi}\int_{0}^{2}(cos^{-1}(\frac{y}{2}))^{2}dy=2{\pi}({\pi}-2)
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