Write an integral expression for the volume of an inverted (upside-down) cone with height 5 and base radius 2. Use <em>h</em> (the distance from the base to a slice) as the variable of integration.
We slice the cone horizontally. The slices are circular with thickness Δ h.
When we cut the cone down the center to look at the similar triangles, we notice that the height of the smaller triangle is (5 – h), not h.
Using the similar triangles and letting x be the radius of the slice, we have
The area of the circle is
and the volume of the slice is
The volume of the entire cone is