Answer
Let S be the solid obtained by rotating the region shown in the
figure about the y-axis. Explain why it is awkward to use slicing
to find the volume V of S. Sketch a typical approximating shell.
What are its circumference and height? Use shells to find V.
Circumference= 2$\pi$x, height= x(x-1)$^{2}$,V= $\pi$/15
Work Step by Step
If we are to conclude that V=2$\pi$rh$\Delta$r meaning V= [circumference][height][thickness], we can find that V$_{i}$ = (2$\pi$x$_{i}$)(f(x$_{i}$))$\Delta$x.
Doing so would require us to take the mean of function and of x. This would prove to be difficult and would be much easier to use the integral function instead.
V=2$\pi$rh$\Delta$r
V=∫$^{1}_{0}$ 2$\pi$x(x$^{3}$-2x$^{2}$+x)dx
V=∫$^{1}_{0}$ 2$\pi$x(x$^{4}$-2x$^{3}$+x$^{2}$)dx
V=2$\pi$[$\frac{x^{5}}{5}$-$\frac{x^{4}}{2}$+$\frac{x^{3}}{3}$]$^{1}_{0}$
V=2$\pi$($\frac{1}{5}$-$\frac{1}{2}$+$\frac{1}{3}$)
V=2$\pi$($\frac{6}{30}$-$\frac{15}{30}$+$\frac{10}{30}$)
V=2$\pi$($\frac{1}{30}$)
V=$\frac{\pi}{15}$
