
Recall our definition of the definite integral of a function of a single variable:
We can extend this definition to define the integral of a function of two or more variables.
Double Integral of a Function of Two VariablesLet $f(x,y)$ be defined on a closed and bounded region $R$ of the $xy$plane. Set up a grid of vertical and horizontal lines in the $xy$plane to form an inner partition of $R$ into $n$ rectangular subregions $R_{k}$ of area $\Delta A_{k}$, each of which lies entirely in $R$. (Ignore the rectangles that are not entirely contained in $R$) Choose a point $(x_{k}^{*}, y_{k}^{*})$ in each subregion $R_{k}$. The sum $$ \sum_{k=1}^{n} f(x_{k}^{*}, y_{k}^{*}) \Delta A_{k} $$ is called a Riemann Sum. In the limit as we make our grid more and more dense, we define the double integral of $f(x,y)$ over $R$ as $$ \iint\limits_R f(x,y\,)dA= \lim_{\max \Delta A_{k} \rightarrow 0} \sum_{k=1}^{n} f(x_{k}^{*}, y_{k}^{*}) \Delta A_{k} . $$ Notes
Geometric Interpretation of the Double IntegralNotice that as we increase the density of our grid, the sum $\sum\limits_{k=1}^{n}A_{k}$ of the individual rectangles better and better approximates the area of region $R$. In the limit as $\Delta A_{k} \rightarrow 0$, we have $$ \mbox{Area of } R = \iint\limits_{R} dA. $$ Suppose now that $f(x,y) \ge 0$ on $R$. Then $f(x_{k}^{*}, y_{k}^{*})\Delta A_{k}$ is the volume of a rectangular parallelopiped of height $f(x_{k}^{*},y_{k}^{*})$ and base area $\Delta A_k$. Adding up these volumes, we get an appoximation for the volume of the solid above $R$ and below the suface $z=f(x,y)$. Thus, in the limit as $\Delta A_{k} \rightarrow 0$,
Note
We next turn to the actual evaluation of double integrals.
Iterated IntegralsIn the double integral $\iint\limits_{R} f(x,y) \,dA$, the differential $dA$ may be viewed informally as an infinitesimal area of a rectangle inside $R$ with dimensions $dy$ and $dx$. For the kinds of "ordinary" functions and regions we'll be concerned with, \begin{eqnarray*} \iint\limits_{R} f(x,y)dA & = & \int_{a}^{b}\left[ \int_{g_{1}(x)}^{g_{2}(x)} f(x,y\,)dy \right] \,dx = \int_{a}^{b}\int_{g_{1}(x)}^{g_{2}(x)} f(x,y\,)dy\,dx \\ & = & \int_{c}^{d} \left[ \int_{h_{1}(y)}^{h_{2}(y)} f(x,y) \,dx \right] \,dy = \int_{c}^{d}\int_{h_{1}(y)}^{h_{2}(y)} f(x,y)\, dx\,dy \end{eqnarray*} where the limits of integration are determined by the region $R$ over which we are integrating. Notes
An example will make these ideas more concrete.
ExampleLet's evaluate the double integral $\displaystyle \iint\limits_{R} 6xy\, dA$, where $R$ is the region bounded by $y=0$, $x=2$, and $y=x^{2}$. We will verify here that the order of integration is unimportant:
so $\displaystyle \iint\limits_{R} 6xy \, dA =32$ here, regardless of the order in which we carry out the integration, as long as we are careful to set up the limits of integration correctly. Now for a triple integral...
ExampleWe will evaluate the triple integral $\displaystyle \int_{0}^{2}\int_{1}^{y^{2}}\int_{1}^{z} yz \, dx \,dz\,dy.$
Key Concepts
We evaluate $\displaystyle\iint\limits_{R} f(x,y) \, dA$ as an iterated integral: \begin{eqnarray*} \iint\limits_{R} f(x,y) \, dA & = & \int_{a}^{b}\int_{g_{1}(x)}^{g_{2}(x)} f(x,y\,)dy\,dx \\ & = & \int_{c}^{d}\int_{h_{1}(y)}^{h_{2}(y)} f(x,y)\, dx\,dy \end{eqnarray*} for "ordinary" regions $R$ and functions $f(x,y)$. 