Answer
$a)$ $\frac{11\pi}{15}$
$b)$ $\frac{97\pi}{105}$
$c)$ $\frac{121\pi}{210}$
$d)$ $\frac{23\pi}{30}$
Work Step by Step
a) the x axis (use shell method)
$V$ = $\int_{{\,0}}^{{\,1}}$$2\pi(y)(1-y+y^{3})$$dy$
$V$ = $2\pi$$\int_{{\,0}}^{{\,1}}$$(y-y^{2}+y^{4})$$dy$
$V$ = $2\pi$$(\frac{1}{2}y^{2}-\frac{1}{3}y^{3}+\frac{1}{5}y^{5})$$|_{{\,0}}^{{\,1}}$
$V$ = $2\pi$$[(\frac{1}{2}-\frac{1}{3}+\frac{1}{5})-(0-0+0)]$
$V$ = $\frac{11\pi}{15}$
b) the y axis (use washer method)
$V$ = $\int_{{\,0}}^{{\,1}}$$\pi[1^{2}-((y-y^{3})^{2}]$$dy$
$V$ = $\pi$$\int_{{\,0}}^{{\,1}}$$[1-y^{2}+2y^{4}-y^{6}]$$dy$
$V$ = $\pi$$(y-\frac{1}{3}y^{3}+\frac{2}{5}y^{5}-\frac{1}{7}y^{7})$$|_{{\,0}}^{{\,1}}$
$V$ = $\pi$$[(1-\frac{1}{3}+\frac{2}{5}-\frac{1}{7})-(0-0+0-0)]$
$V$ = $\frac{97\pi}{105}$
c) the line x= 1 (use washer method)
$V$ = $\int_{{\,0}}^{{\,1}}$$\pi[(1-y+y^{3})^{2}-(1-1)^{2}]$$dy$
$V$ = $\pi$$\int_{{\,0}}^{{\,1}}$$(1-2y+y^{2}+2y^{3}-2y^{4}+y^{6})$$dy$
$V$ = $\pi$$(y-y^{2}+\frac{1}{3}y^{3}+\frac{1}{2}y^{4}-\frac{2}{5}y^{5}+\frac{1}{7}y^{7})$$|_{{\,0}}^{{\,1}}$
$V$ = $\pi$$[(1-1+\frac{1}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{7})-(0)]$
$V$ = $\frac{121\pi}{210}$
d) the line y = 1 (use shell method)
$V$ = $\int_{{\,0}}^{{\,1}}$$2\pi(1-y)(1-y+y^{3})$$dy$
$V$ = $2\pi$$\int_{{\,0}}^{{\,1}}$$(1-2y+y^{2}+y^{3}-y^{4})$$dy$
$V$ = $2\pi$$(y-y^{2}+\frac{1}{3}y^{3}+\frac{1}{4}y^{4}-\frac{1}{5}y^{5})$$|_{{\,0}}^{{\,1}}$
$V$ = $2\pi$$[(1-1+\frac{1}{3}+\frac{1}{4}-\frac{1}{5})-(0)]$
$V$ = $\frac{23\pi}{30}$
