# What is the volume of the solid produced by revolving f(x)=tanx, x in [0,pi/4] around the x-axis?

Apr 13, 2017

The volume is $= 0.67 {u}^{3}$

#### Explanation:

We need
${\sin}^{2} x + {\cos}^{2} x = 1$

Dividing by ${\cos}^{2} x$

${\tan}^{2} x + 1 = {\sec}^{2} x$

${\tan}^{2} x = {\sec}^{2} x - 1$

and

$\int {\sec}^{2} x \mathrm{dx} = \tan x$

A small volume is

$\mathrm{dV} = \pi {y}^{2} \mathrm{dx}$

So,

$V = \pi {\int}_{0}^{\frac{\pi}{4}} {\tan}^{2} x \mathrm{dx}$

$= \pi {\int}_{0}^{\frac{\pi}{4}} \left({\sec}^{2} x - 1\right) \mathrm{dx}$

$= \pi {\left[\tan x - x\right]}_{0}^{\frac{\pi}{4}}$

$= \pi \left(1 - \frac{\pi}{4} - 0\right)$

$= 0.67$