Units

# Unit 7: Double Integrals in Polar Coordinates

## The Concept

Now we will look at the concept of double integrals in polar coordinates. Rather than using a cartesian (or rectangular) coordinate system as we have used thus far to evaluate single and double integrals, we will use the polar coordinate system. The polar coordinate system is a 2D coordinate system in which each point on a plane is determined using a distance from a reference point and an angle from a reference direction. The rectangular coordinate system is best suited for graphs and regions that are naturally considered over a rectangular grid. The polar coordinate system is an alternative that offers good options for functions and domains that have more circular characteristics.

While a point $P$ in rectangular coordinates is described by an ordered pair $(x,y)$, it may also be described in polar coordinates by $(r, \theta)$, where r is the distance from $P$ to the origin and $\theta$ is the angle formed by the line segment and the positive $x$x-axis. We may convert a point from rectangular to polar coordinates using the following equations:

$r =\sqrt{x^2+y^2}$  and $\tan(\theta) = \frac{y}{x}$,

or convert a point from polar to rectangular coordinates using the following equations:

$x =r \cos\theta$  and $y = r \sin\theta$.

The double integral $\iint_D f(x,y)\,dA$ in rectangular coordinates can be converted to a double integral in polar coordinates as $\iint_D f(r \cos\theta, r \sin\theta)\,r\,dr d\theta$.

## The Plot

Now, you should engage with the plot below to understand polar coordinates. Follow the steps below to apply changes to the plot and observe the effects:

1. Change the bounds on the double integral in polar coordinates for both the $r$ and $\theta$ bounds. The bounded region will be shown in the plot and $t$ in the plot represents $\theta$.
2. The result of the double integral in polar coordinates will be shown too.

## Self-Checking Questions

Check your understanding by solving the following questions:

1. $\iint_{D} 3x \, dA$ where $R={(r,\theta)| 0 \leq r \leq 1, 0 \leq \theta \leq 2}$.
2. $\iint_{D} 1-x^2-y^2 \, dA$ where $R={(r,\theta)| 0 \leq r \leq 1, 0 \leq \theta \leq 2\pi}$.

Use the graph to find the answers to these questions. 