A triple integral sums tiny boxes throughout a solid. Integrating 1 gives volume; integrating a density gives mass.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Three nested integrals
The same rule as before applies, one level deeper. Work from the innermost outward.
The limits
The innermost limits may depend on both outer variables. The outermost limits must always be plain numbers.
Mass from density
If density varies through a solid, mass is the triple integral of the density function over the region.
Cylindrical coordinates
Polar coordinates in the plane plus an ordinary z. The area element gains the same factor: r dz dr dθ.
Choosing coordinates
Cylinders and cones favour cylindrical coordinates. Boxes favour rectangular ones. The region decides, not the integrand alone.
Three integrals, one quantity
A triple integral sums over a solid region. With integrand 1 it gives volume; with a density function it gives mass. The integrand determines what the answer means.
Describing a solid with limits
The innermost limits may depend on both outer variables. Setting them up requires understanding the solid, and it is far harder than the integration that follows.
Cylindrical and spherical coordinates
Cylindrical suits solids with an axis of symmetry, spherical those with a centre. Each carries its own volume element — r dz dr dθ and ρ² sin φ dρ dφ dθ — which must not be omitted.
Mass, centre of mass, moment of inertia
All three are triple integrals differing only in the integrand. Once the region is described, computing a family of physical quantities costs little extra work.
Step 2: Try It Yourself
Tap and try it out.
- The surfacez = a·√(x² + y²)
- Height at the origin0
Each line holds one coordinate fixed and varies the other, so every line is a cross-section of the surface.
Step 3: Watch an Example
One step at a time.
Watch Nikhil Find a Mass
Nikhil finds the mass of a 2 by 3 by 4 box whose density is a constant 5.
- Step 1
He recognises that constant density lets the density come outside the integral.
Step 4: Your Turn
Practice makes it stick.
The Block
Problem 1 of 2
A block 3 by 4 by 5 with constant density 2. What is its mass?
The Volume
Problem 2 of 2
A triple integral of the constant 1 over a solid gives 45. What is the volume?
Fill the Solid
1 of 8
A box 2 by 2 by 2. What is its volume?
2 of 8
A solid of volume 12 with constant density 7. What is its mass?
3 of 8
How many nested integrals does a triple integral have?
4 of 8
Must the outermost limits be plain numbers? 1 yes, 0 no.
5 of 8
The cylindrical volume element carries r to what power?
6 of 8
A cylinder of radius 2 and height 5. What is its volume divided by π?
7 of 8
Match each integral to what it computes.
Tap a card on the left to start.
8 of 8
A cube of side 4. What is its volume?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A solid of volume 9 with constant density 6. What is its mass?
Question 2 of 2
What does integrating the constant 1 over a solid give?
What You Learned
- 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.