# Chapter 6 - Section 6.2 - Volume - 6.2 Exercises - Page 447: 32

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}

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