6.2 Volumes/27

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} R &= x^\frac{1}{2} or \sqrt{x} \\[1ex] r &= x^3 \\ \end{align} }

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