
We integrate over regions in polar coordinates.

We are currently interested in computing integrals of functions over various regions in $\R ^2$ and $\R ^3$ via Some regions like rectangles and boxes are easy to describe using $(x,y)$-coordinates (a.k.a. rectangular coordinates). However, other regions like circles and other things with rotational symmetry are easier to work with in polar coordinates. Recall that in polar coordinates,

where $r(\theta )$ is a function of $\theta$. When working with parametric equations of this form, it is common to notate and state that we are working in polar coordinates.

Consider the point $(5, 2\pi /3)$ in polar coordinates. What is this point when expressed in $(x,y)$-coordinates?
Consider the point $(-1, -5)$ in $(x,y)$-coordinates. What is this point when expressed in polar coordinates with $0\le \theta <2\pi$?

Double integrals in polar coordinates

The basic form of the double integral is:

which can be interpreted as

Over some region, sum up products of heights and areas.

Of course if you want to evaluate the integral (and honestly, who doesn’t?) you have to change $R$ to a region defined in $(x,y)$-coordinates, and change $\d A$ to $\d x\d y$ or $\d y\d x$ leaving iterated integrals

Now consider representing a region $R$ with polar coordinates.

Let $R$ be the region in the first quadrant bounded by the curve. We can approximate this region using the natural shape of polar coordinates: Portions of sectors of circles. In the figure, one such region is shaded, shown below:

From the picture above, we see that:

So to evaluate replace $\d A$ with $r\d r\d \theta$ and convert the function $z=F(x,y)$ to a function of polar coordinates: Finally, find bounds $g_1(\theta )\leq r\leq g_2(\theta )$ and $\alpha \leq \theta \leq \beta$ that describe $R$. Let’s state this as a theorem:

Write down a double integral in polar coordinates that will compute the area of a circle of radius $a$.

Finally, let’s derive the volume of a sphere using a double integral in polar coordinates.

One may wonder how polar coordinates could be extended to triple integrals…read on!