Let $f : D \to \mathbb{R}$ be bounded on a bounded region $D \subseteq \mathbb{R}^2$. The double integral of $f$ over $D$ is $\iint_D f(x, y) \, dA = \lim_{\|P\| \to 0} \sum_{k=1}^{n} f(x_k^*, y_k^*) \, \Delta A_k$ where $P$ is a partition of $D$ into subrectangles of area $\Delta A_k$ and $(x_k^*, y_k^*)$ is a sample point in the $k$-th subrectangle. The integral exists when $f$ is continuous on $D$ (or more generally, bounded with discontinuities on a set of measure zero).

Iterated integrals reduce a double integral to successive single integrals. For a region $D = \{(x,y) : a \leq x \leq b, \; g_1(x) \leq y \leq g_2(x)\}$: $\iint_D f(x,y)\,dA = \int_a^b \left(\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\right) dx$

The triple integral of $f$ over a region $E \subseteq \mathbb{R}^3$ is $\iiint_E f(x,y,z)\,dV = \lim_{\|P\| \to 0} \sum_k f(x_k^*, y_k^*, z_k^*)\,\Delta V_k$ and is computed as an iterated integral. For $E = \{(x,y,z) : (x,y) \in D,\; h_1(x,y) \leq z \leq h_2(x,y)\}$: $\iiint_E f\,dV = \iint_D \left(\int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z)\,dz\right) dA$