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### Double Integrals

The same principles apply when integrating in higher dimensions: chop and add. For example, to find the amount of chocolate on a rectangular wafer given its (surface) mass density $\sigma$, first chop the plate into small rectangular pieces, as indicated symbolically in Figure 1. How big are the pieces? Surely \begin{equation} dA = dx \,dy = dy\,dx \end{equation}

What is the mass of each such piece? Clearly, $\sigma \,dA$. Since $\sigma$ is not constant, the total mass is given by \begin{equation} M = \int \sigma \,dA \end{equation} Since we are working in rectangular coordinates, we can write $\sigma=\sigma(x,y)$, so that the total mass is now 1) \begin{equation} M = \dint \sigma(x,y) \,dx\,dy \end{equation}

This is a *double integral*. How does one evaluate such an integral? Not surprisingly, one integrates twice.

Consider the simplest case, when $\sigma$ is constant; if $\sigma=1$, this is equivalent to finding the area of the plate. Work from the inside out; this is an *iterated single integral*. Suppose one corner of the plate is at the origin, and the other at the point $(a,b)$ in the first quadrant. Then \begin{equation} M = \Int_{\hbox{plate}} \!\!\!dA = \Int_0^a \left( \Int_0^b \,dy\right) \,dx = \Int_0^a b \,dx = ab \end{equation} The integrals themselves are easy, but be careful to get the order of integration — and the limits — right. And the answer is, of course, no surprise.

This integral corresponds to first slicing the rectangle into vertical lines of width $dx$, then breaking up each line into segments of length $dy$. One way to indicate this order is to draw a typical line in the region, as shown in Figure 2. The integration is performed in the reverse order: First add up the segments (integrate with respect to $y$), then the lines (integrate with respect to $x$). It is up to you to decide which way to chop first; the answer is the same either way.

In a case such as this one, where both the limits and the integrand are constant, the order simply doesn't matter. More generally, *so long as
the limits are constant*, integrals of the form $\int f(x)\,g(y) \,dA$ can be factored, and the two pieces evaluated independently. For example, \begin{equation} \Int_2^3 \Int_1^2 xy \,dy\,dx = \Int_2^3 x\,dx \Int_1^2 y \,dy = \left(\frac{ x^2}{2}\right)\Bigg|_2^3 \left(\frac{ y^2}{2}\right)\Bigg|_1^2 = \frac{15}{4} \end{equation}