Cylindrical Shells Help

Hello, Yogi!

Quote:

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

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

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