13.2 Evaluation of triple integrals over rectangular and general regions

Triple integrals extend double integrals into three dimensions, letting you compute volumes, masses, and other properties of 3D regions. Setting them up correctly requires understanding the geometry of the region and choosing the right coordinate system and integration order.

Rectangular and General Regions

Defining Rectangular and General Regions

A rectangular region (also called a box region) has constant limits on every variable. This is the simplest case because the bounds don't depend on each other, so you can integrate in any order.

$\int_{a}^{b} \int_{c}^{d} \int_{e}^{f} f(x,y,z) \, dz \, dy \, dx$

Here $a, b, c, d, e, f$ are all constants. The region is literally a rectangular box in 3D space.

A general region has at least some limits that are functions of the other variables. This means the shape of the region changes as you move along one or more axes, so the order of integration matters.

$\int_{a}^{b} \int_{g(x)}^{h(x)} \int_{p(x,y)}^{q(x,y)} f(x,y,z) \, dz \, dy \, dx$

The key rule: a variable's limits can only depend on variables that appear in outer integrals. In the expression above, the $z$-limits can depend on $x$ and $y$ (both integrated outside), and the $y$-limits can depend on $x$ (integrated outermost), but the $x$-limits must be constants.

Classifying General Regions

General regions are classified by which variable has the "dependent" bounds. The classification determines which variable you should integrate first (innermost).

• Type 1 (z-simple): The region is bounded above and below by surfaces $z = q(x,y)$ and $z = p(x,y)$. You integrate $z$ first, then project the region onto the $xy$-plane to find the remaining bounds.

$\int_{a}^{b} \int_{g(x)}^{h(x)} \int_{p(x,y)}^{q(x,y)} f(x,y,z) \, dz \, dy \, dx$

• Type 2 (y-simple): The region is bounded by surfaces $y = g(x,z)$ and $y = h(x,z)$. You integrate $y$ first, then project onto the $xz$-plane.

$\int_{a}^{b} \int_{e}^{f} \int_{g(x,z)}^{h(x,z)} f(x,y,z) \, dy \, dz \, dx$

• Type 3 (x-simple): The region is bounded by surfaces $x = g(y,z)$ and $x = h(y,z)$. You integrate $x$ first, then project onto the $yz$-plane.

$\int_{c}^{d} \int_{e}^{f} \int_{g(y,z)}^{h(y,z)} f(x,y,z) \, dx \, dz \, dy$

Many regions can be described as more than one type. Choosing the right type often makes the integral much easier to evaluate.

Coordinate Systems

Cylindrical Coordinates

Cylindrical coordinates $(r, \theta, z)$ are the natural choice when the region has symmetry around the $z$-axis (think: cylinders, cones, paraboloids centered on the $z$-axis).

• $r$ is the distance from the $z$-axis in the $xy$-plane ($r \geq 0$)
• $\theta$ is the angle in the $xy$-plane, measured counterclockwise from the positive $x$-axis
• $z$ is the same vertical coordinate as in rectangular

Conversion formulas:

• $x = r \cos \theta$
• $y = r \sin \theta$
• $z = z$

The volume element picks up a factor of $r$:

$dV = r \, dr \, d\theta \, dz$

That extra $r$ comes from the Jacobian of the transformation and must always be included.

Spherical Coordinates

Spherical coordinates $(\rho, \theta, \phi)$ work best for regions with symmetry about the origin (spheres, cones, hemispheres).

• $\rho$ is the distance from the origin ($\rho \geq 0$)
• $\theta$ is the same azimuthal angle as in cylindrical (angle in the $xy$-plane)
• $\phi$ is the polar angle measured down from the positive $z$-axis ($0 \leq \phi \leq \pi$)

Conversion formulas:

• $x = \rho \sin \phi \cos \theta$
• $y = \rho \sin \phi \sin \theta$
• $z = \rho \cos \phi$

The volume element is:

$dV = \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi$

The factor $\rho^2 \sin \phi$ is the Jacobian for this transformation. Forgetting it is one of the most common mistakes on exams.

Integration Techniques

Jacobian and Change of Variables

The Jacobian is the determinant of the matrix of partial derivatives for a coordinate transformation. It tells you how volume elements scale when you switch coordinate systems.

For a transformation $(x,y,z) \to (u,v,w)$, the change of variables theorem says:

$\iiint_D f(x,y,z) \, dx \, dy \, dz = \iiint_E f(x(u,v,w),\, y(u,v,w),\, z(u,v,w)) \, \left| \frac{\partial(x,y,z)}{\partial(u,v,w)} \right| \, du \, dv \, dw$

The absolute value of the Jacobian determinant replaces the volume element. For the standard coordinate systems:

• Cylindrical: Jacobian = $r$, so $dV = r \, dr \, d\theta \, dz$
• Spherical: Jacobian = $\rho^2 \sin \phi$, so $dV = \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi$

You can also define custom transformations for regions that don't fit standard systems. The process is the same: compute the Jacobian matrix, take its determinant, and multiply the integrand by the absolute value.

Determining Bounds of Integration

Setting up the correct bounds is usually the hardest part of a triple integral. Here's a systematic approach:

1. Sketch the region (or at least identify its bounding surfaces).
2. Choose a coordinate system based on the geometry. Cylindrical symmetry suggests cylindrical coordinates; spherical symmetry suggests spherical.
3. Pick the innermost variable and find its bounds in terms of the remaining variables by solving for where the region starts and ends along that direction.
4. Project the region onto the plane of the two remaining variables. This projection gives you a 2D region to set up as a double integral.
5. Find the bounds for the middle variable in terms of the outermost variable.
6. The outermost variable always has constant bounds.

Example: For a sphere of radius $a$ centered at the origin, spherical coordinates give constant bounds on every variable:

• $0 \leq \rho \leq a$
• $0 \leq \theta \leq 2\pi$
• $0 \leq \phi \leq \pi$

In rectangular coordinates, the same sphere would require variable bounds ($z$ depends on $x$ and $y$, etc.), making the integral much harder. This is exactly why choosing the right coordinate system matters so much.