# How do you find the volume of the solid #y=-x+1# revolved about the x-axis?

##### 1 Answer

If we restrict

about the x-axis, and its volume will be

#### Explanation:

If you were to revolve the entire function

Let's instead choose to consider only the function on the interval

(the solid of revolution will be a cone)

Picture the integral by chopping the cone into a series of infinitely thin vertical slices.

Each slice is then a cylinder with its radius equal to

So the volume of one slice is

and the volume of the entire solid is the sum (integral) of all slices

In this problem,

Now let's find the definite integral from 0 to 1.

So the volume of the cone obtained by revolving

about the x-axis is

Note: This agrees with the formula for the volume of a cone

Our cone has radius of 1 and a height of 1, so