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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)
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.
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)