Answer
a.) $V=3.70110$
b.) $V=6.16850$
Work Step by Step
a.) $\displaystyle{A(x)=\pi\left(\cos^2x\right)^2}\\ \displaystyle{A(x)=\pi\left(\cos^4x\right)}$
$\begin{aligned} V &=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} A(x) \ d x \\ V &=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \pi\left(\cos^4x\right) \ d x \\ V &=\pi \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^4x\ dx \\ V &=2\pi \int_{0}^{\frac{\pi}{2}} \cos^4x\ dx \\ V&=3.70110 \end{aligned}$
b.) $\displaystyle{A(x)=\pi\left(1-0\right)^2-\pi\left(1-\cos^2x\right)^2}\\ \displaystyle{A(x)=\pi\left(2\cos^2x-\cos^4x\right)}$
$\begin{aligned} V &=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} A(x) \ d x \\ V &=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \pi\left(2\cos^2x-\cos^4x\right) \ d x \\ V &=\pi \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 2\cos^2x-\cos^4x\ dx \\ V &=2\pi \int_{0}^{\frac{\pi}{2}} 2\cos^2x-\cos^4x\ dx \\ V&=6.16850 \end{aligned}$
