Find the volume of the solid generated by revolving the region in the first quadrant bounded by the $x$-axis and the curve $y = x \cos x , 0 \leq x \leq \pi / 2$ about the $y$-axis.
A) $\frac { \pi ^ { 3 } } { 2 } - 4 \pi$
B) $\frac { \pi ^ { 3 } } { 2 } + 2 \pi ^ { 2 } - 4 \pi$
C) $\frac { \pi ^ { 2 } } { 2 } - 4 \pi$
D) $\frac { \pi ^ { 3 } } { 2 } - 8 \pi$

Question

Quizplus · Accepted Answer

The volume of the solid generated by revolving the region bounded by the curve $y = x \cos x$ for $0 \leq x \leq \pi / 2$ about the y-axis can be found using the method of cylindrical shells. The formula for the volume of a shell is $V = 2\pi \int_{a}^{b} x f(x) dx$, where $f(x) = x \cos x$ in this case. Integrating from $0$ to $\pi/2$, we get $V = 2\pi \int_{0}^{\pi/2} x^2 \cos x dx$. Solving this integral gives us $\frac{\pi^3}{2} - 4\pi$, which matches option A.

Find the Volume of the Solid Generated by Revolving the Region