The Main Idea 

Some regions and functions are naturally suited to polar coordinates rather than rectangular coordinates. Converting to polar coordinates can transform a difficult integral into a manageable one, especially for circular or radial regions.

A polar rectangular region is bounded by constant [latex]r[/latex]-values and constant [latex]\theta[/latex]-values:

[latex]R = {(r,\theta) \mid a \leq r \leq b, \alpha \leq \theta \leq \beta}[/latex]

This describes a sector of an annulus (ring) with inner radius [latex]a[/latex], outer radius [latex]b[/latex], and angular span from [latex]\alpha[/latex] to [latex]\beta[/latex].

The Critical Formula

When converting to polar coordinates, the area element changes:

[latex]dA = r\,dr\,d\theta[/latex]

The extra [latex]r[/latex] factor is essential—it accounts for how polar “rectangles” get wider as you move away from the origin. Never forget this [latex]r[/latex].

Double Integral in Polar Form

[latex]\underset{R}{\displaystyle\iint}f(r,\theta)dA = \displaystyle\int_{\theta=\alpha}^{\theta=\beta}\displaystyle\int_{r=a}^{r=b}f(r,\theta)r\,dr\,d\theta[/latex]

Converting from Rectangular

When [latex]f[/latex] is given in [latex]x[/latex] and [latex]y[/latex], use the conversions:

• [latex]x = r\cos\theta[/latex]
• [latex]y = r\sin\theta[/latex]
• [latex]dA = r,dr,d\theta[/latex]

For example, [latex]x^2 + y^2[/latex] becomes simply [latex]r^2[/latex].

Problem-Solving Strategy

1. Identify if polar is appropriate: Look for circles, sectors, or expressions involving [latex]x^2 + y^2[/latex]
2. Determine bounds: Find the [latex]r[/latex]-range (radii) and [latex]\theta[/latex]-range (angles) that describe your region
3. Convert the integrand: Replace [latex]x[/latex] and [latex]y[/latex] with [latex]r\cos\theta[/latex] and [latex]r\sin\theta[/latex]
4. Don’t forget the [latex]r[/latex]: Include it in [latex]r\,dr\,d\theta[/latex]
5. Integrate: Usually integrate with respect to [latex]r[/latex] first, then [latex]\theta[/latex]

Use polar coordinates when:

• The region is circular, semicircular, or a sector
• The integrand contains [latex]x^2 + y^2[/latex] (which becomes [latex]r^2[/latex])
• Boundaries are easier to express as [latex]r = \text{constant}[/latex] or [latex]\theta = \text{constant}[/latex]

The Main Idea 

Just as rectangular regions generalize beyond rectangles to curves, polar regions extend beyond simple sectors. For regions with curved polar boundaries, you integrate with variable limits that depend on [latex]\theta[/latex].

General Polar Regions

A general polar region has the form:

[latex]R = {(r,\theta) \mid \alpha \leq \theta \leq \beta, h_1(\theta) \leq r \leq h_2(\theta)}[/latex]

The angle [latex]\theta[/latex] ranges between constants, while the radius [latex]r[/latex] varies between two functions of [latex]\theta[/latex]. Think of it as sweeping an angle from [latex]\alpha[/latex] to [latex]\beta[/latex], where at each angle the radius extends from inner curve [latex]h_1(\theta)[/latex] to outer curve [latex]h_2(\theta)[/latex].

The Iterated Integral

[latex]\underset{R}{\displaystyle\iint}f(r,\theta)dA = \displaystyle\int_{\theta=\alpha}^{\theta=\beta}\displaystyle\int_{r=h_1(\theta)}^{r=h_2(\theta)}f(r,\theta)r\,dr\,d\theta[/latex]

The outer integral spans the angular range (constants). The inner integral spans the radial range (functions of [latex]\theta[/latex]).

Problem-Solving Strategy

1. Sketch the region: Draw the polar curves to visualize the boundaries
2. Find the angular range: Determine [latex]\alpha[/latex] and [latex]\beta[/latex] where the region exists
3. Identify radial bounds: At each angle [latex]\theta[/latex], find where [latex]r[/latex] starts ([latex]h_1(\theta)[/latex]) and ends ([latex]h_2(\theta)[/latex])
4. Set up the integral: Place angle limits outside, radial limits (as functions) inside
5. Include the [latex]r[/latex]: Always multiply by [latex]r[/latex] in [latex]r\,dr\,d\theta[/latex]
6. Integrate: Work inside out—first with respect to [latex]r[/latex], treating [latex]\theta[/latex] as constant

Key Reminder

When radial bounds depend on [latex]\theta[/latex], you’re describing how far out the region extends at each particular angle. This is the polar equivalent of Type I regions in rectangular coordinates.

The Main Idea 

Polar coordinates excel at computing volumes and areas for circular or radial regions. The same double integral framework applies, but the geometry often makes calculations dramatically simpler than rectangular coordinates.

Volume Under a Surface

For a solid under [latex]z = f(r,\theta)[/latex] over a polar region [latex]D[/latex]:

[latex]V = \underset{D}{\displaystyle\iint}f(r,\theta)r\,dr\,d\theta[/latex]

Convert [latex]x^2 + y^2[/latex] to [latex]r^2[/latex] in the function, and don’t forget the [latex]r[/latex] in [latex]r,dr,d\theta[/latex].

Area of a Polar Region

To find the area of region [latex]D[/latex], integrate 1:

[latex]A = \displaystyle\int_\alpha^\beta\displaystyle\int_{h_1(\theta)}^{h_2(\theta)}r\,dr\,d\theta[/latex]

The [latex]r[/latex] factor handles the “spreading out” of area elements as radius increases.

Strategic Decisions

Not every problem is easier in polar coordinates. If the region is triangular or the function doesn’t simplify with [latex]r^2 = x^2 + y^2[/latex], stick with rectangular coordinates. Compare the complexity of both setups before committing.

Problem-Solving Strategy

1. Sketch the region: Identify whether it’s naturally circular, radial, or involves [latex]x^2 + y^2[/latex]
2. Find polar bounds: Determine [latex]\alpha \leq \theta \leq \beta[/latex] and [latex]h_1(\theta) \leq r \leq h_2(\theta)[/latex]
3. Convert the function: Replace [latex]x[/latex] and [latex]y[/latex] with [latex]r\cos\theta[/latex] and [latex]r\sin\theta[/latex]; simplify [latex]x^2 + y^2[/latex] to [latex]r^2[/latex]
4. Set up carefully: Include the [latex]r[/latex] in [latex]r,dr,d\theta[/latex]—this is non-negotiable
5. Integrate: Usually [latex]r[/latex] first, then [latex]\theta[/latex]

Common Applications

• Cones: For [latex]z = h - \frac{h}{a}\sqrt{x^2 + y^2}[/latex] over a circle, convert to [latex]z = h - \frac{h}{a}r[/latex]—much simpler.
• Polar roses: For [latex]r = \cos(n\theta)[/latex], use symmetry. Find the area of one petal and multiply by the number of petals.
• Intersecting regions: When a circle and cardioid overlap, split the integral at intersection points, integrating different radial bounds over different angular ranges.
• Improper Integrals: The entire plane becomes [latex]0 \leq \theta \leq 2\pi[/latex], [latex]0 \leq r < \infty[/latex], requiring limits as [latex]r \to \infty[/latex].