Chapter 7: Transcendental Functions - Section 7.6 - Inverse Trigonometric Functions - Exercises 7.6 - Page 422: 108

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

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