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Conceptually, computing double integrals in polar coordinates is the same as in rectangular coordinates. After all, the idea of an integral doesn't depend on the coordinate system. If $R$ is a region in the plane and $f(x,y)$ is a function, then $\iint_R f(x,y) dA$ is what we get when we
The differences when working with polar coordinates are explained in the following video:
So what is the area of a polar rectangle? The polar rectangle is the difference of two pie wedges, one with radius $b$ and one with radius $a$, and so has area \begin{eqnarray*} \Delta A & = & \frac{b^2}{2}(\beta\alpha)  \frac{a^2}{2}(\beta  \alpha) \\ & = & \frac{b^2a^2}{2}(\beta\alpha) \\ & = & \frac{a+b}{2} (ba)(\beta  \alpha) \\ & = & r^* \Delta r \Delta \theta, \end{eqnarray*} where $r^*=\frac{a+b}{2}$ is the average value of $r$, $\Delta r = ba$ is the change in $r$ and $\Delta \theta = \beta\alpha$ is the change in $\theta$. Another way to understand this is to look at the shape of a small polar rectangle, like the shaded box in the figure above. This is almost a rectangular, only rotated and with slightly curved sides. One side has length $\Delta r$. The other had length approximately $r \Delta \theta$. So the whole rectangle has approximate area $r \Delta r \Delta \theta$.
