Chapter 6 - Section 6.2 - Volumes Using Cylindrical Shells - Exercises - Page 365: 46

$\displaystyle \pi(\frac{4\pi}{3}-\tan\frac{\pi}{3})\approx 7.718$

Apply the washer method (rotation is about the y-axis) $V=\displaystyle \int_{c}^{d}\pi[(R(y))^{2}-(r(y))^{2}]dy$ $R(y)=2,\quad r(y)=\sec y.$ $V=\displaystyle \pi\int_{0}^{\pi/3}[2^{2}-\sec^{2}y]dy$ $=\pi[4y-\tan y]_{0}^{\pi/3}$ $=\displaystyle \pi(\frac{4\pi}{3}-\tan\frac{\pi}{3})\approx 7.718$