Evaluate the triple integral B
xyz2 dV,
where B is the rectangular box
B =

(x, y, z) | 0 ≤ x ≤ 1, −1 ≤ y ≤ 7, 0 ≤ z ≤ 4

,
integrating first with respect to y, then z, and then x.

Each face of the box B is described completely by a single variable, so the value of the integral is independent of the order of integration.

The integral in question would be

but we can factorize the integral as

We can do this regardless of the order of integration we choose. Then

and the triple integral has a value of

But, if you insist on carrying out just one integral at a time, we have

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