Chapter 5 - Applications of Integration - 5.3 Volumes by Cylindrical Shells - 5.3 Exercises - Page 382: 3

$\displaystyle{V=\frac{6\pi}{7}}$

$\displaystyle{V=\int_{0}^{1}(2\pi x)\left(\sqrt[3]x\right)\ dx}\\ \displaystyle{V=\int_{0}^{1}(2\pi x)\left(x^{\frac{1}{3}}\right)\ dx}\\ \displaystyle{V=2\pi\int_{0}^{1}x^{\frac{4}{3}}\ dx}\\ \displaystyle{V=2\pi\left[\frac{3}{7}x^{\frac{7}{3}}\right]_{0}^{1}}\\ \displaystyle{V=2\pi\left(\left(\frac{3}{7}(1)^{\frac{7}{3}}\right)-(0)\right)}\\ \displaystyle{V=\frac{6\pi}{7}}$