Answer
$a)$ $\frac{8\pi}{3}$
$b)$ $\frac{8\pi}{5}$
$c)$ $8\pi$
$d)$ $4\pi$
Work Step by Step
a) the x axis
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(y)(\frac{y^{2}}{2}-\frac{y^{4}}{4}+\frac{y^{2}}{2})$$dy$
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(y)(y^{2}-\frac{y^{4}}{4})$$dy$
$V$ = $2\pi$$\int_{{\,0}}^{{\,2}}$$(y^{3}-\frac{y^{5}}{4})$$dy$
$V$ = $2\pi$$(\frac{x^{4}}{4}-\frac{y^{6}}{24})$$|_{{\,0}}^{{\,2}}$
$V$ = $2\pi$$[(4-\frac{8}{3})-(0-0)]$
$V$ = $\frac{8\pi}{3}$
b) the line y = 2
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(2-y)(\frac{y^{2}}{2}-\frac{y^{4}}{4}+\frac{y^{2}}{2})$$dy$
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(2-y)(y^{2}-\frac{y^{4}}{4})$$dy$
$V$ = $2\pi$$\int_{{\,0}}^{{\,2}}$$2y^{2}-y^{3}-\frac{y^{4}}{2}+\frac{y^{5}}{4})$$dy$
$V$ = $2\pi$$(\frac{2}{3}y^{3}-\frac{1}{4}y^{4}-\frac{1}{10}{y^{5}}+\frac{1}{24}{y^{6}})$$|_{{\,0}}^{{\,2}}$
$V$ = $2\pi$$[(\frac{16}{3}-4-\frac{16}{5}+\frac{8}{3})-(0-0-0+0)]$
$V$ = $\frac{8\pi}{5}$
c) the line y = 5
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(5-y)(\frac{y^{2}}{2}-\frac{y^{4}}{4}+\frac{y^{2}}{2})$$dy$
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(5-y)(y^{2}-\frac{y^{4}}{4})$$dy$
$V$ = $2\pi$$\int_{{\,0}}^{{\,2}}$$5y^{2}-y^{3}-\frac{5}{4}{y^{4}}+\frac{y^{5}}{4})$$dy$
$V$ = $2\pi$$(\frac{5}{3}y^{3}-\frac{1}{4}y^{4}-\frac{1}{4}{y^{5}}+\frac{1}{24}{y^{6}})$$|_{{\,0}}^{{\,2}}$
$V$ = $2\pi$$[(\frac{40}{3}-4-8+\frac{8}{3})-(0-0-0+0)]$
$V$ = $8\pi$
d) the line y = $-\frac{5}{8}$
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(y+\frac{5}{8})(\frac{y^{2}}{2}-\frac{y^{4}}{4}+\frac{y^{2}}{2})$$dy$
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(y+\frac{5}{8})(y^{2}-\frac{y^{4}}{4})$$dy$
$V$ = $2\pi$$\int_{{\,0}}^{{\,2}}$$\frac{5}{8}y^{2}+y^{3}-\frac{5}{32}{y^{4}}-\frac{y^{5}}{4})$$dy$
$V$ = $2\pi$$(\frac{5}{24}y^{3}+\frac{1}{4}y^{4}-\frac{1}{32}{y^{5}}-\frac{1}{24}{y^{6}})$$|_{{\,0}}^{{\,2}}$
$V$ = $2\pi$$[(\frac{5}{3}+4-1-\frac{8}{3})-(0-0-0+0)]$
$V$ = $4\pi$
