6.2 Volumes/27

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

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} 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{align} }