Chapter 6 - Section 6.1 - Volumes Using Cross-Sections - Exercises - Page 356: 34

$4$

We need to integrate the integral to compute the volume. We have: $$Area =\pi r^2=\pi (\sqrt {\cos (\dfrac{\pi y}{4}})^2=\pi \cos (\dfrac{\pi y}{4})$$ $$Volume = \pi \int_{-2}^{0} \cos (\dfrac{\pi y}{4}) \space dy \\= \pi [(4 \pi) \sin (\dfrac{\pi y}{4})]_{-2}^{0} \\=4 [\sin (0) -\sin (\dfrac{-\pi}{2})] \\ =4$$