How do you find the volume of the solid #y=-x+1# revolved about the x-axis?
If we restrict
about the x-axis, and its volume will be
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