1. Math 200
Week 9 - Wednesday
Triple Integrals

2. Math 200
Goals
Be able to set up and evaluate triple integrals using rectangular, cylindrical, and spherical coordinates

3. Math 200
Triple Integrals
We integrate functions of three variables over three dimensional solids S dV
(x0,y0,z0)
Chop the solid S up into a bunch of cubes with volume dV
Pick a point in each cube and evaluate F there
Add up all of these products (F•dV)

4. Math 200
Interpretations
If we think of F(x,y,z) as giving the density of the solid S at (x,y,z), then the triple integral gives us the mass of S
If F(x,y,z) = 1, then the integral gives us the volume of S dV S
(x0,y0,z0)

5. Math 200
Example 1

6. Math 200
Lots of ways to setup
Let’s set up a few triple integrals for the volume of the solid bounded by y2 + z2 = 1 and y = x in the first octant
This means, we’ll just integrate F(x,y,z) = 1
Here’s what the solid looks like:
A sketch will really help with these problems

7. Let’s say we want to integrate in the order dzdydx
Math 200
Let’s say we want to integrate in the order dzdydx
Once we integrate with respect to z, z is gone
Visually, we can think of flattening the solid onto the xy-plane
The top bound for z is the surface z2=(1-y2)1/2
The bottom bound is the xy- plane, z1=0

8. We have y1=x & y2=1 and x1=0 & x2=1
Math 200
Once we’ve flattened out in the z-direction, we have a double integral to set up, which we already know how to do!
We have y1=x & y2=1 and x1=0 & x2=1
So the triple integral becomes

9. Alternatively, we could have gone with dzdxdy
Math 200
Alternatively, we could have gone with dzdxdy
In this case all that changes is the outer double integral
Going back to the flattened image on the xy-plane, we get x1=0 & x2=y and y1=0 & y2=1

10. We could also not start with z. For example, let’s try dxdzdy
Math 200
We could also not start with z. For example, let’s try dxdzdy
Integrating with respect to x first will flatten the picture onto the yz-plane
On “top” (meaning further out towards us), we have x2=y
On the “bottom” (meaning further back) we have x1=0
yz-plane

11. So the triple integral becomes
Math 200
Now we just set up the bounds for the outer two integrals based on the flattened image on the yz- plane
z1=0 & z2=(1-y2)1/2
y1=0 & y2=1
So the triple integral becomes

12. Pick one of these three to integrate:
Math 200
Pick one of these three to integrate:
With the dzdydx integral, we end up needing trig substitution to perform the second integration, so we should go with the second or third option

13. Math 200

14. Math 200
Example 2

15. Flatten the solid onto the xy- plane
Math 200
Let’s try dzdydx first
z1=0 and z2=y
Flatten the solid onto the xy- plane
Now for y we have…
y1=x2 & y2=4
Finally,
x1=-2 & x2=2

16. We have to be careful when using symmetry:
Math 200
We have to be careful when using symmetry:
It works here because both the function we’re integrating and the region over which we’re integrating are symmetric over the plane x=0.

17. Let’s look at some alternative setups
Math 200
Let’s look at some alternative setups
We could have started with x instead and done dxdzdy
If we draw a line through the solid in the x-direction, it first hits the back half of the parabolic surface and then the front half of the parabolic surface
If we then collapse the picture onto the yz-plane, we get this…

18. Triple integrals in Cylindrical and spherical Coordinates
Math 200
Triple integrals in Cylindrical and spherical Coordinates
Cylindrical coordinates
We already know from polar that dA = rdrdθ
So, for dV we get rdrdθdz
Just replace dxdy or dydx with rdrdθ
Spherical coordinates
For now, let’s just accept that in spherical coordinates, dV becomes ρ2sinφdρdφdθ
We’ll come back to why this is the case in the next section

19. EXAMPLE Consider the integral
Math 200
EXAMPLE
Consider the integral
First let’s get a sense of what the region/solid looks like
z1 = -(4 - x2 - y2)1/2 and z2 = (4 - x2 - y2)1/2
Squaring both sides of either equation, we get a sphere of radius 2 centered at (0,0,0): x2 + y2 + z2 = 4
So we’re going from the bottom half of the sphere to the top half
y1 = 0 & y2 = (4-x2)1/2 and x1 = 0 & x2 = 2
On the xy-plane, we go from the line y=0 to the top half of a circle of radius 2, but only from x=0 to x=2.

20. Math 200

21. Setup in cylindrical coordinates
Math 200
Setup in cylindrical coordinates
Since z is common to rectangular and cylindrical, let’s start with that
Now we can look at what remains on the xy-plane and convert that to polar (recall: cylindrical = polar + z)
y1 = 0 & y2 = (4-x2)1/2 and x1 = 0 & x2 = 2
On the xy-plane, we go from the line y=0 to the top half of a circle of radius 2, but only from x=0 to x=2.
r goes from 0 to 2 and θ goes from 0 to π/2

22. Math 200

23. For spherical, let’s start with ρ:
Math 200
For spherical, let’s start with ρ:
The sphere of radius 2 is simply ρ=2
The region starts at the origin: ρ=0
Remember, φ measures the angle taken from the positive z- axis
In order to cover the quarter-sphere, φ needs to go from 0 to π.
We already know what θ does from cylindrical coordinates
The integrand (the function we’re integrating) is a little more involved…

24. Lastly, we can’t forget about the “extra term,” ρ2sinφ:
Math 200
Lastly, we can’t forget about the “extra term,” ρ2sinφ: