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One Variable at the TimeNow that we know what double integrals are, we can start to compute them. The key idea is: One variable at a time! In order to integrate over a rectangle $[a,b] \times [c,d]$, we first integrate over one variable (say, $y$) for each fixed value of $x$. That's an ordinary integral, which we can do using the fundamental theorem of calculus. We then integrate the result over the other variable (in this case $x$), which we can also do using the fundamental theorem of calculus. So a 2dimensional double integral boils down to two ordinary 1dimensional integrals, one inside the other. We call this an iterated integral. There are two ways to see the relation between double integrals and iterated integrals. In the bottomup approach, we evaluate the sum $$\sum_{i=1}^m \sum_{j=1}^n f\left(x_{i}^*,y_{j}^*\right) \,\Delta x\, \Delta y,$$ by first summing over all of the boxes with a fixed $i$ to get the contribution of a column, and then adding up the columns. (Or we can sum over all of the boxes with a fixed $j$ to get the contribution of a row, and then add up the rows.)
This bottomup approach is explained in the following video. (Video Fix? However, there is a small error. At the beginning it says that we're going to integrate over the rectangle $[0,1] \times [0,2]$, but for the rest of the video the region $R$ is actually the rectangle $[0,2] \times [0,1]$.) Cavalieri's PrincipleAn alternate approach to finding volumes (and hence double integrals)
 the socalled Slice Method  was formulated by
Cavalieri
and is expressed mathematically in
We already used this idea to compute volumes of revolution. Suppose $W$ is created by rotating the graph of $y = f(x),\, a \le x \le b,$ about the $x$axis. When $P_x$ is a plane perpendicular to the $x$axis, then the slice of $W$ cut by $P_x$ is a disk of radius $f(x)$. Here $A(x) = \pi f(x)^2$, so we recover the familiar result $$\hbox{ volume of} \ W \ = \ \pi \int_a^b\, f(x)^2\, dx$$ for a volume of revolution. But Cavalieri's Principle does not require the crosssections to be triangles or disks!
In other words, the volume of a region is $\int_a^b A(x)\, dx$, where $A(x)$ is the crosssectional area at a particular value of $x$. But that's the area under the curve $z=f(x,y)$, where we are treating $x$ as a constant and $y$ as our variable. That is,
