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\pi\}[/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.
