We integrate over regions in polar coordinates.

*rotational*symmetry are easier to work with in polar coordinates. Recall that in polar coordinates,

where is a function of . When working with parametric equations of this form, it is
common to notate and state that we are working in *polar coordinates*.

meaning:

**polar coordinates**.

### Double integrals in polar coordinates

The basic form of the double integral is:

**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 to a region defined in -coordinates, and change to or leaving iterated integrals

Now consider representing a region with polar coordinates.

Let 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:

and computing:

So to evaluate replace with and convert the function to a function of polar coordinates: Finally, find bounds and that describe . Let’s state this as a theorem:

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

The region we need to integrate over is the circle of radius , centered at the origin. Thus, the volume of a sphere with radius is:

The formula for the volume of a sphere with radius is given as . We have justified this formula with our calculation!

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