Welcome to the jungle. We won't find any scrumptious sweets here, unless our jungle is in the middle of Oz. Our well-being depends on us being able to tackle finding volumes of more complicated objects, including pyramids, cones, and spheres.
If we just stand here helpless, we are vulnerable to an attack by a swarm of killer bees. We are going to jump right in to our voyage and see if we can integrate some of these shapes.
Sample Problem
Find an integral expression for the volume of a pyramid whose height is 7 and whose base is square with sides of length 5.
Answer.
The pyramid looks like:
If we slice the pyramid horizontally, we get a square slice:
We can use h as the variable of integration, where h measures the distance from the top of the pyramid to the slice. Then the thickness of the slice is Δ h. To find the side-length of the square, we can cut the pyramid straight down the middle from top to bottom and behold the similar triangles.
Let's have x be the length of the side of the square. Using similar triangles,
so
.
The area of the square is
so the volume of the slice at height h is
The volume of the pyramid is
To find the volume of a cone, we do the same thing except that the slices are now circles instead of squares. We use similar triangles to find the radius of each slice. The only difference is that we'd rather eat ice cream out of a cone instead of a pyramid.
Finally we come to the spheres. This is shaped like that baseball your threw at your brother's head when you were five. All these years later, he's still mad about it. And this calculus section is karma's way of repaying the favor.
It turns out that spheres just aren't that complicated. Whether we slice a sphere horizontally or vertically we get circular slices.
First, we need an equation for these circles. The equation of a sphere centered at the origin is
x2 + y2 + z2 = r2.
If we set any one of those variables to 0, the remaining variables satisfy the equation for a circle. For example, if z = 0 then
x2 + y2 = r2.
Practice:
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, we want to 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 we would have to do a more complicated integral involving a square root. If we had to solve this integral right now, karma and your brother would be getting the last laugh. Since we don't, we just managed to set up a nice integral to look at and have a calculator solve. | |
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 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
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:
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.
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.
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
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. The radius of the slice is 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 in front.
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 slice the sphere horizontally, with h 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.
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 will 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.
