Cogito
Multivariable Calculus · Chapter 7 · Lesson 3
Triple Integrals
Three integrals, and the answer is mass or volume.
12 problems · about 24 minutes · F-IF.B.6, G-GMD.A.3
What this lesson teaches
The student sets up triple integrals for volume and mass over solid regions.
- A triple integral sums tiny boxes throughout a solid region.
- Integrating 1 gives volume; integrating a density gives mass.
- Cylindrical coordinates carry the same extra factor of r as polar coordinates.
Warm Up
Straightforward practice. Get the method working first.
5 problemsA solid of volume 9 with constant density 6. What is its mass?
Answer 54
Why 54.
What does integrating the constant 1 over a solid give?
Answer The volume of the solid.
Why Volume.
A box 2 by 2 by 2. What is its volume?
Answer 8
Why 2 cubed.
A solid of volume 12 with constant density 7. What is its mass?
Answer 84
Why Multiply.
How many nested integrals does a triple integral have?
Answer 3
Why The name says it.
Build It Up
The same ideas with more to keep track of.
3 problemsMust the outermost limits be plain numbers? 1 yes, 0 no.
Answer 1
Why Nothing is left to depend on.
The cylindrical volume element carries r to what power?
Answer 1
Why The same single r as polar.
A cylinder of radius 2 and height 5. What is its volume divided by π?
Answer 20
Why 4 × 5.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each integral to what it computes.
Answer Triple integral of 1 → Volume of a solid; Triple integral of a density → Mass of a solid; Double integral of 1 → Area of a region
Why Integrating 1 always returns the size of the region.
A cube of side 4. What is its volume?
Answer 64
Why 4 cubed.
The Block: A block 3 by 4 by 5 with constant density 2. What is its mass?
Answer 120
Why 120.
The Volume: A triple integral of the constant 1 over a solid gives 45. What is the volume?
Answer 45
Why 45.