Write an integral expression for the volume of a cone with height 8 and base radius 5.
The cone we want to integrate looks like:
Let h be the distance from the tip of the cone to the slice. A slice will be a circle with thickness Δ h.
If we cut the cone down the middle, we can see the similar triangles. The distance from the middle of the slice to the edge is the radius of the slice. We can call this radius x.
Since this is the radius of the circular slice, the area of the circle is
and the volume of the slice is
The variable h goes from 0 (at the tip of the cone) to 8 (at the base of the cone), so the volume of the entire cone is
We promised no food, but we came across some ice cream. We better eat it quick before a bear smells it.