Answer
$V=2\pi \int_{0}^{\pi/3} x(\tan x-x)dx $
Work Step by Step
Given
$$y=\tan x, y=x, x=\pi / 3 ; \quad \text { about the } y \text {-axis }$$
To find the volume of the solid when the bounded region rotates about $y-axsis$, we use the method of shell
\begin{aligned}
V&= 2\pi \int_a^b r(x)h(x)dx
\end{aligned}
Here
$$r(x)= x,\ \ h(x) =\tan x-x $$
Then
\begin{aligned}
V&= 2\pi \int_a^b r(x)h(x)dx\\
&= 2\pi \int_{0}^{\pi/3} x(\tan x-x)dx
\end{aligned}
