## Calculus: Early Transcendentals 8th Edition

$2 \pi \int _{0}^{\frac{ \pi }{3}}x \left( tanx-x \right) dx$
{Step 1 of 2} Consider the region enclosed by the graphs of $y=tanx$, $y=x$ and $x=\frac{ \pi }{3}.$ {Step 2 of 2} We use the method of cylindrical shells.A typical shell is shown above.The radius of the shell is r(x)=x and the height of the shell is $h \left( x \right) =tanx-x$ . The volume V of the resulting solid of revolution is given by $V=2 \pi \int _{0}^{\frac{ \pi }{3}}r \left( x \right) h \left( x \right) dx$ $=2 \pi \int _{0}^{\frac{ \pi }{3}}x \left( tanx-x \right) dx$