Triple Integrals over General Domains

# Triple Integrals over General Domains

Let $w = f(x, y, z)$ be a three variable real-valued function. Like with double integrals, sometimes the region for which we are integrating over isn't a box but instead some other subset of the domain $D(f)$. We will develop a similar method for evaluating triple integrals over general domains.

First consider a general bounded region $E$ that is a subset of $D(f)$. Since $E$ is a bounded region, then we can enclose $E$ within a box $B$. Define $\hat{f}$ as $\quad \hat{f} (x, y, z) = \left\{\begin{matrix} f(x, y, z) & \mathrm{if (x, y, z) \in E} \\ 0 & \mathrm{if (x, y, z) \not \in E} \end{matrix}\right.$. Then it's not hard to see that:

(1)
\begin{align} \quad \iiint_E f(x, y, z) \: dV = \iiint_B \hat{f}(x, y, z) \: dV \end{align}

Now there are three main types of domains we will integrate over as illustrated in the following diagram.

• Type 1 Regions: The first type of region we might integrate over is in the form $E = \{ (x, y, z) : (x, y) \in D, u_1(x, y) ≤ z ≤ u_2(x, y) \}$. We are thus integrating $f$ for which $(x, y) \in D$ and $z$ is trapped between the surfaces generated by $u_1$ and $u_2$. Thus we have that for type 1 regions:
(2)
\begin{align} \quad \iiint_E f(x, y, z) \: dV = \iint_D \left [ \int_{u_1(x, y)}^{u_2(x, u)} f(x, y, z) \: dz \right ] \: dA \end{align}
• Type 2 Regions: The second type of region we might integrate over is in the form $E = \{ (x, y, z) : (y, z) \in D, u_1(y, z) ≤ x ≤ u_2(y, z) \}$. We are thus integrating $f$ for which $(y, z) \in D$ and $x$ is trapped between the surfaces generated by $u_1$ and $u_2$. Thus we have that for type 2 regions:
(3)
\begin{align} \quad \iiint_E f(x, y, z) \: dV = \iint_D \left [ \int_{u_1(y, z)}^{u_2(y, z)} f(x, y, z) \: dx \right ] \: dA \end{align}
• Type 3 Regions: The third type of region we might integrate over is in the form $E = \{ (x, y, z) : (x, z) \in D, u_1(x, z) ≤ y ≤ u_2(x, z) \}$. We are thus integrating $f$ for which $(x, z) \in D$ and $y$ is trapped between the surfaces generated by $u_1$ and $u_2$. Thus we have that for type 3 regions:
(4)
\begin{align} \quad \iiint_E f(x, y, z) \: dV = \iint_D \left [ \int_{u_1(x, z)}^{u_2(x, z)} f(x, y, z) \: dy \right ] \: dA \end{align}

After reducing either case to a double integral, we will then need to evaluate the succeeding double integral which we've already seen how to do.