Fubini's Theorem for Evaluating Triple Integrals over Boxes

# Fubini's Theorem for Evaluating Triple Integrals over Boxes

Recall from the Fubini's Theorem and Evaluating Double Integrals over Rectangles page that if $z = f(x, y)$ was a two variable real-valued function for which $f$ was continuous on the rectangle $R = [a, b] \times [c,d]$ then the double integral of $f$ over $R$ could be evaluated as iterated integrals and:

(1)
\begin{align} \quad \iint_R f(x, y) \: dA = \int_c^d \int_a^b f(x, y) \: dx \: dy \end{align}

We will now look at an analogous Theorem for evaluating triple integrals over a rectangular box $B$ with iterated integrals.

 Theorem 1: Let $w = f(x, y, z)$ be a three variable real-valued function such that $f$ is continuous on the rectangular box $B = [a, b] \times [c, d] \times [r, s]$. Then the triple integral of $f$ over $B$ can be evaluated as iterated integrals, that is, $\iiint_B f(x, y, z) \: dV = \int_r^s \int_c^d \int_a^b f(x, y, z) \: dx \: dy \: dz$.

There are six total ways to evaluate a triple integral over a box using iterated integrals. For example, we could get that $\iiint_B f(x, y, z) \: dV = \int_a^b \int_c^d \int_r^s f(x, y, z) \: dz \: dy \: dx$, or $\iiint_B f(x, y, z) \: dV = \int_c^d \int_a^b \int_r^s f(x, y, z) \: dz \: dx \: dy$. Each of these six possible orders will give rise to the same value.

Let's now look at an example of evaluating a triple integral over a box.

## Example 1

Evaluate the triple integral $\iiint_B 2x - e^y + 3z^2 \: dV$ over the box $B = [0, 2] \times [0, 1] \times [1, 3]$.

By Fubini's Theorem we can rewrite this triple integral as iterated integrals:

(2)
\begin{align} \quad \iiint_B 2x - e^y + 3z^2 \: dV = \int_1^3 \int_0^1 \int_0^2 2x - e^y + 3z^2 \: dx \: dy \: dz \\ \quad \iiint_B 2x - e^y + 3z^2 \: dV = \int_1^3 \int_0^1 \left [x^2 - xe^y + 3xz^2 \right ]_0^2 \: dy \: dz \\ \quad \iiint_B 2x - e^y + 3z^2 \: dV = \int_1^3 \int_0^1 4 - 2e^y + 6z^2 \: dy \: dz \\ \quad \iiint_B 2x - e^y + 3z^2 \: dV = \int_1^3 \left [ 4y - 2e^y + 6yz^2 \right ]_0^1 \: dz \\ \quad \iiint_B 2x - e^y + 3z^2 \: dV = \int_1^3 6 - 2e + 6z^2 \: dz \\ \quad \iiint_B 2x - e^y + 3z^2 \: dV = \left [ 6z - 2ez + 2z^3 \right]_1^3 \\ \quad \iiint_B 2x - e^y + 3z^2 \: dV = (18 - 6e + 54) - (6 - 2e + 2) \\ \quad \iiint_B 2x - e^y + 3z^2 \: dV = 64-4e \end{align}