Triple Integrals over General Regions

Triple integrals over general bounded regions extend the concept of double integrals from planar domains to three-dimensional solids. A solid region E in space is commonly enclosed within a rectangular box B, and a continuous function f(x, y, z) is integrated over the region by defining F such that it coincides with f on E and is zero outside the solid. The triple integral is therefore expressed as

\begin{equation*}\iiint_E f(x,y,z) dV \end{equation*}

The existence of the integral requires that f be continuous and that the boundary of E be sufficiently smooth. Similar to double integrals, the evaluation of triple integrals depends strongly on the geometric description of the region.

A common class of solids is the type I region, where the variable z is bounded between two continuous surfaces:

\begin{equation*}E=\{(x,y,z)\mid (x,y)\in D; u_1(x,y)\le z\le u_2(x,y)\}\end{equation*}

Here, D represents the projection of the solid onto the xy-plane. The lower boundary of the solid is given by z = u₁(x, y), while the upper boundary is described by z = u₂(x, y). In this case, the triple integral can be rewritten as an iterated integral:

\begin{equation*}\iiint_E f(x,y,z)\, dV=\iint_D\left[\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z)\, dz\right] dA\end{equation*}

If the projection D is a type I plane region,

\begin{equation*}a\le x\le b,\qquad g_1(x)\le y\le g_2(x)\end{equation*}

The integral becomes

\begin{equation*}\int_a^b\int_{g_1(x)}^{g_2(x)}\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z) dz\, dy\, dx\end{equation*}

Alternatively, when D is a type II plane region described by

\begin{equation*}c\le y\le d,\qquad h_1(y)\le x\le h_2(y)\end{equation*}

The order of integration changes accordingly:

\begin{equation*}\int_c^d\int_{h_1(y)}^{h_2(y)}\int_{u_1(x,y)}^{u_2(x,y)}f(x,y,z) dz\, dx\, dy\end{equation*}

These representations provide flexible methods for evaluating integrals over complex three-dimensional solids.