# What is the surface area produced by rotating f(x)=abs(1-x), x in [0,3] around the x-axis?

##### 1 Answer
Jan 23, 2017

$5 \sqrt{2} \setminus \pi$

#### Explanation:

If we look at the graph $y = \left\mid 1 - x \right\mid$ we get:
graph{|1-x| [-10, 10, -5, 5]}

so rotating about )x will produce two cones:

One smaller cone of height $1$, and one larger cone of height $3$,

The surface area of a cone of radius $r$ is given by $\pi r l$ where $l$ is the length of the slope:

For the smaller cone:

${l}^{2} = {1}^{2} + {1}^{2} \implies l = \sqrt{2}$

For the larger cone:

${l}^{2} = {2}^{2} + {2}^{2} \implies l = \sqrt{8} = 2 \sqrt{2}$

So total surface area is:

$S A = \left(\pi\right) \left(1\right) \left(\sqrt{2}\right) + \left(\pi\right) \left(2\right) \left(2 \sqrt{2}\right)$
$\setminus \setminus \setminus \setminus \setminus = 5 \sqrt{2} \setminus \pi$