Questions: Triple Integrals in Cartesian Coordinates

Question 1

The region W is defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 4 − x² − y². A student evaluates the innermost integral ∫₀^(4−x²−y²) dz. What does the result of this step represent?

• A. The total volume of W, computed in a single step
• B. The value 4 − x² − y², a function of x and y that represents the height of the solid at each point (x,y), still to be integrated over the base region
• C. A constant equal to the average height of the solid
• D. Zero, because integrating 1 over a symmetric region cancels out

Answer: (B)

When you evaluate the innermost integral ∫₀^(4−x²−y²) 1 dz with x and y held fixed, you get 4 − x² − y² — a function that depends on x and y. This represents the height of the solid at that particular horizontal position. You have not yet computed the full volume; you have computed a height function that still needs to be integrated over the base region (0 ≤ x ≤ 1, 0 ≤ y ≤ 1) to sum up all those heights. The triple integral is completed by the two remaining integrations over y and x.

Question 2

A solid W is described as: 'at each height z from 0 to 2, the cross-section is the disk x² + y² ≤ z².' What is the most natural order of integration for computing ∭_W f dV?

• A. Integrate z first (innermost), then x, then y — because z determines the boundary
• B. Integrate x first, then y, then z — alphabetical order is always simplest
• C. Integrate x and y first (inner and middle), then z last (outer) — because the solid is naturally described as cross-sections at each z
• D. All six orderings are equally simple for this region — choice is arbitrary

Answer: (C)

When a solid is described via cross-sections at each z — 'at height z, the cross-section is ...' — the natural structure calls for integrating over x and y first (for each fixed z) and then integrating over z last. For each fixed z, the cross-section x² + y² ≤ z² is a disk of radius z, which is easily described in polar coordinates. If you tried to integrate z first, you would need to invert the region description — finding for each (x,y) the range of z — which is algebraically more complicated. Choosing the order that matches the natural description of the solid is the core skill in setting up triple integrals.

Question 3

If f(x,y,z) = 1 everywhere in the region W, then ∭_W dV computes the surface area of W.

• T. True
• F. False

Answer: False

When f = 1, the triple integral ∭_W 1 dV computes the volume of W, not the surface area. Each infinitesimal piece dV = dx dy dz contributes its volume to the sum, and integrating over the entire region W totals all these pieces into the full three-dimensional volume. Surface area requires a different kind of integral — a surface integral that accounts for the shape and orientation of the boundary, not an integration over the interior.

Question 4

All six orderings of a triple integral (dx dy dz, dx dz dy, dy dx dz, etc.) give the same numerical answer when computed correctly, even though the limits of integration look different for each ordering.

• T. True
• F. False

Answer: True

This is Fubini's theorem applied to triple integrals: as long as f is continuous (or integrable) over W, the value of ∭_W f dV is independent of the order of integration. What changes between orderings is the description of the limits — some orderings produce simple constant limits, others produce complicated functions. The skill in choosing an order is not about getting a different answer but about finding the algebraically cleanest path to the same answer. A solid that is easy to describe as 'upper surface minus lower surface over a base region' naturally suggests integrating z first.

Question 5

Why is choosing the order of integration a skill in setting up triple integrals, rather than an arbitrary convention? What factors determine which ordering is most practical?

Model answer: The six orderings all give the same value, but they require different limit expressions. The order that matches the natural geometric description of the solid produces the simplest limits. If the solid is described as 'lying between two surfaces for (x,y) in a base region,' integrating the bounded variable (z) first produces simple function-valued limits for z and constant limits for x and y. If instead you integrate in the wrong order, the limits of the inner integral may be impossible to express simply, or the resulting integrals may be much harder to evaluate analytically.

In practice, the first question to ask is: how is the boundary of W described most naturally? If it is 'above the plane z=0 and below the paraboloid z = 4−x²−y²,' then z is the natural innermost variable. If it is 'inside the cylinder x²+y² ≤ 1 from z=0 to z=3,' then x and y are the natural inner variables with z outer. Drawing the solid and identifying which variable's bounds depend on the others directs the order choice. Some regions require switching to cylindrical or spherical coordinates to become tractable, which is the motivation for those coordinate systems.