Answer
$a)$ $8\pi$
$b)$ $\frac{32\pi}{5}$
$c)$ $\frac{224\pi}{15}$
$d)$ $\frac{8\pi}{3}$
Work Step by Step
use shell method
a) the x axis
$y$ = $\sqrt x$
$y^{2}$ = $x$
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(y)(y^{2})$$dy$
$V$ = $2\pi$$\int_{{\,0}}^{{\,2}}$$(y^{3})$$dy$
$V$ = $2\pi$$(\frac{1}{4}y^{4})$$|_{{\,0}}^{{\,2}}$
$V$ = $2\pi$$(4-0)$ = $8\pi$
b) the y axis
$V$ = $\int_{{\,0}}^{{\,4}}$$2\pi(x)(2-\sqrt x)$$dx$
$V$ = $2\pi$$\int_{{\,0}}^{{\,4}}$$(2x-x^{\frac{3}{2}})$$dx$
$V$ = $2\pi$$(x^{2}-\frac{2}{5}x^{\frac{5}{2}})$$|_{{\,0}}^{{\,4}}$
$V$ = $2\pi$$[(16-\frac{64}{5})-(0-0)]$
$V$ = $\frac{32\pi}{5}$
c) the line x = 4
$V$ = $\int_{{\,0}}^{{\,4}}$$2\pi(4-x)(2-\sqrt x)$$dx$
$V$ = $2\pi$$\int_{{\,0}}^{{\,4}}$$(8-4\sqrt x-2x+x^{\frac{3}{2}})$$dx$
$V$ = $2\pi$$(8x-\frac{8}{3}x^{\frac{3}{2}}-x^{2}+\frac{2}{5}x^{\frac{5}{2}})$$|_{{\,0}}^{{\,4}}$
$V$ = $2\pi$$[(32-\frac{64}{3}-16+\frac{64}{5})-(0-0-0+0)]$
$V$ = $\frac{224\pi}{15}$
d) the line y = 1
$V$ = $\int_{{\,0}}^{{\,2}}$$2\pi(y-1)(y^{2})$$dy$
$V$ = $2\pi$$\int_{{\,0}}^{{\,2}}$$(y^{3}-y^{2})$$dy$
$V$ = $2\pi$$(\frac{1}{4}y^{4}-\frac{1}{3}y^{3})$$|_{{\,0}}^{{\,2}}$
$V$ = $2\pi$$[(4-\frac{8}{3})-(0-0)]$
$V$ = $\frac{8\pi}{3}$
