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Math101learn.math101.caTriple Integrals
A rigorous, example-driven guide to triple integrals, including hypotheses, method choice, verification, and practice.
The central idea
For an integrable scalar field $f$ on a solid $E\subset\mathbb R^3$, $\iiint_Ef\,dV$ is the limit of weighted volume sums. With $f=1$ it gives volume; with density it gives mass. Fubini's theorem permits iterated integration when appropriate integrability hypotheses hold, and bounds must describe each point of $E$ exactly once.
Definitions, hypotheses, and notation
A solid can be simple in one order and require several pieces in another. Start from the innermost coordinate as a line segment between two surfaces, then project onto a planar region already understood from double integrals. In Cartesian coordinates the Jacobian is one; cylindrical and spherical systems alter it.
Mass moments multiply density by coordinate distances, and centroids divide first moments by total mass. Units provide a strong distinction: density times $dV$ gives mass, while an additional length factor gives a first moment. Symmetry can force some moments to zero before calculation.
Conceptual meaning
A triple integral accumulates through three-dimensional cells. The innermost integral compresses a line of cells, the next combines a sheet, and the outermost completes the solid. Coordinate changes reshape cells and introduce a Jacobian.
A dependable method and decision rule
- Sketch the solid and its projection onto a coordinate plane.
- Choose an order and write inner lower/upper surfaces.
- Describe the projected planar region with the outer bounds.
- Integrate one variable at a time, holding outer variables fixed.
- Check volume, units, symmetry, and whether bounds cover the solid once.
Fully worked example
Graphical or geometric meaning
At fixed $(x,y)$ in the triangular projection, a vertical segment rises from $z=0$ to the slanted plane $z=1-x-y$. These segments shrink to zero along the projection's hypotenuse.
Common mistakes and why they fail
Verification and reasonableness checks
- Set density to one and compare with geometric volume.
- Inspect cross-sections at boundary values.
- Reverse order or use symmetry in a simple case.
Bounds should reconstruct the solid
A triple integral accumulates density over volume. In an iterated description, the innermost variable ranges between two surfaces while the outer variables range over its projection. Sketch or describe that projection first, then identify lower and upper surfaces. Changing order means rebuilding the solid's description and may require splitting. Setting the integrand to one should return the volume; use this as a setup test before integrating a density. Under suitable integrability hypotheses, Fubini's theorem permits iterated evaluation. Keep units visible: density times $dV$ must produce the desired total quantity. Symmetry can eliminate odd contributions only when both the domain and remaining factors possess the necessary symmetry.
Practice
- Find volume of $[0,1]\times[0,2]\times[0,3]$.
- What does $\iiint_E\rho dV$ compute?
- What is the average value of $f$ on $E$?
Answers and brief solutions
- $6$.
- Mass when $\rho$ is volume density.
- $\iiint_Ef dV/\operatorname{Vol}(E)$.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the volume of x,y,z≥0 with x+y+z≤1?
- Integrating z gives height 1−x−y.
- The y integral is (1−x)²/2.
- Integrating x from 0 to 1 gives 1/6.
End of lesson
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