Pyramids, Cones, and Spheres Examples

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.

Then

so

.

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.

Now that we have an equation for a circle, write an integral expression for the volume of a sphere with radius 5.

Let's slice the sphere vertically. Since each slice is a circle, we center the sphere at the origin and use x as the variable of integration. The slices have thickness Δ x. If we cut the sphere in half down the middle at the xz-plane, we can see what half of the sphere and half of a slice look like.

The radius of the slice is z. We cut the sphere along the xz-plane, where y = 0, so x² + z² = 5².

This means

The volume of the slice is then

The variable x goes from -5 to 5, so the volume of the sphere is

Alternately, since the sphere is symmetric we could find half its volume and then multiply by 2.

If we do things this way we get the expression

Notice that since we're squaring that square root, the radical is going to disappear, leaving us with a very easy integral to compute. No calculators required.