6.2 Volumes/27

${\begin{aligned}R&={\sqrt {x}}\\[1ex]r&=x^{3}\\\end{aligned}}$

${\begin{aligned}v=\pi \int _{0}^{1}\left[(x^{\frac {1}{2}})^{2}-(x^{3})^{2}\right]dx&=\pi \int _{0}^{1}\left[(x-x^{6})\right]dx\\[2ex]&=\pi \left[{\frac {x^{2}}{2}}-{\frac {x^{7}}{7}}\right]{\Bigg |}_{0}^{1}\\[2ex]&=\pi \left[{\frac {1}{2}}-{\frac {1}{7}}\right]=\pi \left[{\frac {7-2}{14}}\right]\\[2ex]&={\frac {5\pi }{14}}\end{aligned}}$