Evaluating Triple Integrals over Boxes Examples 1

Evaluating Triple Integrals over Boxes Examples 1

Recall from the Fubini's Theorem for Evaluating Triple Integrals over Boxes page that if $w = f(x, y, z)$ is a three variable real-valued functions and $f$ is continuous over the box $B = [a, b] \times [c, d] \times [r, s]$, then the triple integral of $f$ over the box $B$ can be evaluated as iterated integrals:

(1)
\begin{align} \quad \iiint_B f(x, y, z) \: dV = \int_r^s \int_c^d \int_a^b f(x, y, z) \: dx \: dy \: dz \end{align}

Let's look at some examples of evaluating triple integrals over boxes.

Example 1

Evaluate the triple integral $\iiint_B 2x + 3y + 4z \: dV$ where $B = [0, 1] \times [0, 2] \times [0, 3]$.

We immediately set up our triple integral as iterated integrals and evaluate:

(2)
\begin{align} \quad \iiint_B 2x + 3y + 4z \: dV = \int_0^3 \int_0^2 \int_0^1 2x + 3y + 4z \: dx \: dy \: dz \\ \quad \iiint_B 2x + 3y + 4z \: dV = \int_0^3 \int_0^2 \left [ x^2 + 3xy + 4xz \right ]_{x=0}^{x=1} \: dy \: dz \\ \quad \iiint_B 2x + 3y + 4z \: dV = \int_0^3 \int_0^2 1 + 3y + 4z \: dy \: dz \\ \quad \iiint_B 2x + 3y + 4z \: dV = \int_0^3 \left [ y + \frac{3y^2}{2} + 4yz \right ]_{y=0}^{y=2} \: dz \\ \quad \iiint_B 2x + 3y + 4z \: dV = \int_0^3 8 + 8z \: dz \\ \quad \iiint_B 2x + 3y + 4z \: dV = \left [ 8z + 4z^2 \right ]_{z=0}^{z=3} \\ \quad \iiint_B 2x + 3y + 4z \: dV = 24 + 36 \\ \quad \iiint_B 2x + 3y + 4z \: dV = 60 \end{align}

Example 2

Evaluate the triple integral $\iiint_B xye^x \: dV$ where $B = [0, 1] \times [0, 2] \times [0, 1]$.

We immediately set up our triple integral as iterated integrals and evaluate.

(3)
\begin{align} \quad \iiint_B xye^{x} \: dV = \int_0^1 \int_0^2 \int_0^1 xye^x \: dx \: dy \: dz \\ \quad \iiint_B xye^{x} \: dV = \int_0^1 \int_0^2 y \left [xe^x - e^x \right]_{x=0}^{x=1} \: dy \: dz \\ \quad \iiint_B xye^{x} \: dV = \int_0^1 \int_0^2 y \: dy \: dz \\ \quad \iiint_B xye^{x} \: dV = \int_0^1 \left [ \frac{y^2}{2} \right ]_{0^2} \: dz \\ \quad \iiint_B xye^{x} \: dV = \int_0^1 2 \: dz \\ \quad \iiint_B xye^{x} \: dV = \left [ 2z \right]_{z=0}^{z=1} \\ \quad \iiint_B xye^{x} \: dV = 2 \end{align}