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

[latex]r =\sqrt{x^2+y^2}[/latex] and [latex]\tan(\theta) = \frac{y}{x}[/latex],

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

[latex]x =r \cos\theta[/latex] and [latex]y = r \sin\theta[/latex].

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

## The Plot

Now, you should engage with the plot below to understand polar coordinates^{[1]}. Follow the steps below to apply changes to the plot and observe the effects:

- Change the bounds on the double integral in polar coordinates for both the [latex]r[/latex] and [latex]\theta[/latex] bounds. The bounded region will be shown in the plot and [latex]t[/latex] in the plot represents [latex]\theta[/latex].
- The result of the double integral in polar coordinates will be shown too.

## Self-Checking Questions

Check your understanding by solving the following questions^{[2]}:

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

Use the graph to find the answers to these questions.