Mastering Triple Integrals

Triple integrals extend the concept of double integrals to functions of three variables, allowing us to calculate volumes, masses, and other quantities within three-dimensional regions.

Learning Objectives

After studying this chapter, you should be able to:

• Define the triple integral as a limit of Riemann sums over a 3D domain.
• Evaluate triple integrals over rectangular boxes using Fubini's Theorem.
• Set up iterated integrals for general bounded regions by identifying projections onto coordinate planes.
• Rearrange the six possible orders of integration to simplify complex calculations.
• Calculate physical properties such as volume and average value of a function in space.

Key Terms & Glossary

• Triple Riemann Sum: The sum $\sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(x_{ijk}^*, y_{ijk}^*, z_{ijk}^*) \Delta V$ used to approximate the total value of a function over a 3D region.
• Fubini's Theorem (Triple): A theorem stating that if $f$ is continuous on a rectangular box, the triple integral can be evaluated as an iterated integral in any of the 6 possible orders.
• Iterated Integral: An integral performed sequentially (e.g., first $z$, then $y$, then $x$), where inner variables are treated as constants for the outer integration steps.
• Projection: The 2D shadow ($D$) of a 3D region ($E$) onto one of the coordinate planes (usually $xy, yz, \text{ or } xz$).

The "Big Idea"

If a single integral $\int f(x) dx$ represents the area under a curve and a double integral $\iint f(x,y) dA$ represents the volume under a surface, the triple integral $\iiint f(x,y,z) dV$ represents the "hyper-volume" or the accumulation of a property (like mass or temperature) throughout a solid object. If $f(x,y,z) = 1$, the triple integral simply yields the Volume of the region.

Formula / Concept Box

Hierarchical Outline

1. Integration over Rectangular Boxes
   • Definition: $B = [a, b] \times [c, d] \times [e, f]$.
   • Fubini's Theorem: Order does not matter for continuous functions over boxes.
2. Integration over General Regions
   • Type I Regions: $z$ is bounded by two surfaces $u_1(x,y)$ and $u_2(x,y)$.
   • The Projection $D$: The domain of $(x,y)$ in the $xy$-plane.
3. Evaluation Strategy
   • Inner Limits: Usually involve variables (functions of the outer variables).
   • Outer Limits: Must always be constants.

Visual Anchors

Order of Integration Workflow

3D Coordinate Space Partitioning

Definition-Example Pairs

• Continuity Requirement: A function must be continuous (or bounded with few discontinuities) to be integrable.
  • Example: Calculating the heat distribution in a solid copper block where the temperature function $T(x,y,z)$ varies smoothly.
• Six Possible Orders: The permutations of $dx, dy, dz$.
  • Example: If the region is bounded by $x=y^2$, it might be easier to integrate $dx$ before $dy$ to avoid square roots.

Worked Examples

Example 1: Evaluation over a Box

Problem: Evaluate $\iiint_B xyz^2 \, dV$ where $B = [0,1] \times [-1,2] \times [0,3]$.

Solution:

1. Set up iterated integral: $\int_0^3 \int_{-1}^2 \int_0^1 xyz^2 \, dx \, dy \, dz$
2. Integrate with respect to $x$: $\int_0^3 \int_{-1}^2 [\frac{1}{2}x^2yz^2]_0^1 \, dy \, dz = \int_0^3 \int_{-1}^2 \frac{1}{2}yz^2 \, dy \, dz$
3. Integrate with respect to $y$: $\int_0^3 [\frac{1}{4}y^2z^2]_{-1}^2 \, dz = \int_0^3 (\frac{4}{4}z^2 - \frac{1}{4}z^2) \, dz = \int_0^3 \frac{3}{4}z^2 \, dz$
4. Integrate with respect to $z$: $[\frac{1}{4}z^3]_0^3 = \frac{27}{4} = 6.75$

[!TIP] When the integrand $f(x,y,z)$ is a product of single-variable functions (e.g., $g(x)h(y)k(z)$) and the limits are constant, you can evaluate the triple integral as the product of three separate single integrals.

Checkpoint Questions

1. How many different orders of integration are possible for a triple integral?
2. What is the value of the triple integral $\iiint_E 1 \, dV$ if $E$ is a cube with side length 2?
3. If you project a 3D region onto the $xz$-plane, which differential should be evaluated first in the iterated integral? (e.g., $dx, dy, \text{ or } dz$)?

▶Click to expand answers

1. 6 (xyz, xzy, yxz, yzx, zxy, zyx).
2. 8 (Volume $= 2^3 = 8$).
3. $dy$ (The variable not in the projection plane is integrated first).