Answer
a) $V$ = $\frac{576\pi}{7}$
b) $V$ = $\frac{96\pi}{5}$
Work Step by Step
a) x-ais
disk method
$V$ = $\pi\int_0^2(8-x^{3})^{2}dx$
$V$ = $\pi\int_0^2(64-16x^{3}+x^{6})dx$
$V$ = $\pi(64x-4x^{4}+\frac{1}{7}x^{7})|_0^2$
$V$ = $\frac{576\pi}{7}$
shell method
$V$ = $2\pi\int_0^{8}y(8-y)^{\frac{1}{3}}dy$
let
$u$ = $8-y$
then
$du$ = $-dy$
$V$ = $2\pi\int_0^{8}(8-u)(u)^{\frac{1}{3}}du$
$V$ = $2\pi\int_0^{8}(8u^{\frac{1}{3}}-u^{\frac{4}{3}})du$
$V$ = $2\pi(6u^{\frac{4}{3}}-\frac{3}{7}u^{\frac{7}{3}})|_0^8$
$V$ = $\frac{576\pi}{7}$
b) y axis
disk method
$V$ = $\pi\int_0^8(8-y)^{\frac{2}{3}}dy$
$V$ = $-\frac{3\pi}{5}(8-y)^{\frac{5}{3}}|_0^8$
$V$ = $\frac{96\pi}{5}$
shell method
$V$ = $2\pi\int_0^{2}x(8-x^{3})dx$
$V$ = $2\pi(4x^{2}-\frac{1}{5}x^{5})|_0^2$
$V$ = $\frac{96\pi}{5}$
