Find an integral expression for the volume of a pyramid with height 9 and a square base with side-length 6. Use h as the variable of integration, where h measures the distance from the base of the pyramid to a slice.
Answer
This is very similar to the example. We slice the pyramid horizontally and get a slice that's a square:
We cut the pyramid in half from top to bottom to see the similar triangles. Since h is the distance from the base of the pyramid to the slice, h is no longer the height of the smaller triangle. The height of the smaller triangle is now
(9 – h). If we let x be the length of the side of the slice, we get
so
The area of the square is
so the volume of the slice is
The volume of the entire pyramid is
Example 2
Find an integral expression for the volume of a pyramid with height h and a square base with side-length b. Evaluate the integral. This will give you a formula for the volume of a pyramid.
Hint
Don't use h as the variable of integration since it's already being used as the height of the pyramid. Pick a different letter.
Answer
This problem is like the example except that we don't have numbers anymore. We just have letters. Good grief. Since h is being used as the height of the pyramid, let's use y for the distance from the top of the pyramid to a slice. We will use y as the distance from the top of the pyramid to the slice, rather than the bottom of the pyramid to the slice. It makes the similar triangle ratios nicer. We cut the pyramid down the middle to see the similar triangles. Letting x be the length of the side of the square, we have
so
The area of the square is
and the volume of the slice is
The volume of the pyramid is
Let's evaluate the integral. Remember that b and h are constants, so we can move them outside of the integral:
Example 3
Write an integral expression for the volume of an inverted (upside-down) cone with height 5 and base radius 2. Use h (the distance from the base to a slice) as the variable of integration.
Answer
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, since we're measuring from the tip.
Using the similar triangles and letting x be the radius of the slice, we have
so
The area of the circle is
and the volume of the slice is
The volume of the entire cone is
Example 4
Write an integral expression for the volume of a cone with height h and base radius r. Evaluate your expression to get a formula for the volume of a cone.
Answer
This is just like the example except there aren't any numbers. Since h is being used for the height of the cone, let's use y for the distance from the tip of the cone to the slice. Then the thickness of the slice is Δ y. We cut the cone down the middle to see the similar triangles, and mark the radius of the slice as x. Using similar triangles,
so
The circular side of the slice has area
and the slice has volume
Since the variable y goes from 0 (at the tip of the cone) to h (at the base of the cone), the volume of the cone is
Let's work out the integral. Remember that r, h, and π are constants so we can pull them out of the integral if need be.
Look forward? It should. That's the formula for the volume of a cone.
Example 5
Write an integral expression for the volume of a sphere with radius 4. Use horizontal slices and let h be the depth of the slice below the top of the sphere.
Answer
We'll start by slicing the sphere horizontally, then we'll mark h as the depth of the slice below the top of the sphere. Each slice has thickness Δ h.
If we cut the sphere in half along the xz-plane we can see half the sphere and half the slice. The radius of the slice is x where x2 + z2 = 42. Notice that z = 4 – h.
Since z = 4 – h, the Pythagorean Theorem says
x2 + (4 – h)2 = 42
so
This means the area of the circular side of the slice is
The volume of a slice is
π(42– (4 – h)2) Δ h,
and the volume of the sphere is
We can simplify this integral using the sphere's symmetry to
We could solve either integral, but it's up to you to solve the one that's easiest. We recommend the second one.
Example 6
Write an integral expression for the volume of a sphere with radius r. Evaluate your expression to get a formula for the volume of a sphere.
Answer
This is like the example but with no numbers. Yes, there's a theme here. We told you we would give you the power to derive any area formula you wanted. We'll slice the sphere vertically and use x to denote the position of a slice. Each slice has thickness Δ x. If we cut the sphere down the middle we can see the radius of the slice is z where x2 + z2 = r2. This means so the area of the circle is
and the volume of the slice is
π(r2 – x2) Δ x.
The volume of the sphere is
or
We'll evaluate the second integral. Drats. Karma finally caught up.
There it is—our nice, familiar formula for the volume of a sphere.
so the area of the circle is
