Finding the Area of a Region Bounded by One Polar Curve

What Polar Area Means

When a region is described by a polar equation, its points are represented using radius and angle rather than rectangular coordinates. A polar equation has the form , where tells us how far the point is from the origin at angle .

To find the area enclosed by one polar curve, imagine dividing the region into many very thin sectors. Each small sector has radius and a small angle . Its area is approximately the area of a circular sector:

Adding all those tiny sectors and taking the limit gives the polar-area formula.

The Polar Area Formula

If the polar curve traces a region from to , then the area of that region is:

The factor of is essential. It comes directly from the sector-area formula and should not be omitted.

The Most Important Question: What Are the Bounds?

The formula itself is usually the easy part. On AP Calculus BC problems, the harder step is deciding where the curve begins and ends tracing the region you want.

For a region bounded by one polar curve, the bounds are usually found by determining when the curve returns to the origin or when the desired loop closes.

For example, consider the cardioid:

At , the curve reaches the origin. It then traces the entire cardioid as goes from to . Therefore, the total enclosed area is found using those angular bounds.

A Step-by-Step Method

Write the polar equation. Identify the function .
Determine the interval of . Find the angles over which the curve traces the region exactly once.
Set up the polar-area integral. Use .
Square the radius function. Be especially careful with parentheses: if , then the integrand contains .
Integrate. Use trigonometric identities when necessary.
Check the result. The answer should be positive and should be consistent with the approximate size of the region.

Worked Examples

Example 1: Area Inside a Circle

Find the area enclosed by the polar curve .

Step 1: The curve is a circle centered at with radius . It is traced once as goes from to .

Step 2: Set up the area integral:

Step 3: Evaluate:

Result: The enclosed area is .

Example 2: Area of a Cardioid

Find the area enclosed by the cardioid .

Step 1: The cardioid begins and ends at the origin when . One complete tracing occurs over .

Step 2: Use the polar-area formula:

Step 3: Square the radius:

Step 4: Use the identity :

Step 5: Integrate:

Result: The enclosed area is .

Example 3: A Four-Petaled Rose

Find the area of one petal of the polar curve .

Step 1: One petal is traced from to . These are the consecutive angles at which the curve passes through the origin around the petal centered on the positive -axis.

Step 2: Set up the area:

Step 3: Square the radius:

Step 4: Integrate:

Result: One petal has area .

Example 4: Area of a Region Traced by One Polar Curve

Find the total area enclosed by .

Step 1: The curve is a circle centered at . The radius is .

Step 2: Because the circle is traced once over a full revolution, use to .

Step 3: Set up the integral:

Step 4: Evaluate:

Result: The enclosed area is .

When the Radius Is Negative

A major difference between polar and rectangular coordinates is that can be negative. A negative radius does not mean the curve has negative area. Instead, the point is plotted in the direction opposite the angle.

For area calculations, this is one reason the formula uses . Squaring the radius makes the contribution to area nonnegative.

However, a negative radius can affect which part of the curve is being traced. Always understand the geometry before choosing your limits.

Symmetry Can Make the Integral Easier

Many polar curves have symmetry. If the region is symmetric about the polar axis, you may be able to calculate half the area and multiply by 2.

For example, if a curve satisfies symmetry about the polar axis and the desired region is traced from to , then you can sometimes write:

This is mathematically equivalent to integrating over the entire symmetric interval, but it can make the integral substantially easier.

Common Mistakes

Forgetting the one-half

The polar-area formula is , not . The comes from the area of a circular sector.

Using the wrong bounds

The bounds must correspond to the part of the curve that traces the region you want. Don't automatically use to just because it is one full revolution.

Forgetting to square the entire radius

If , the integrand is . It is not . Put the entire expression for inside parentheses before squaring.

Assuming negative radius means negative area

Polar curves can have negative values of , but area is never negative. The squared radius in the area formula accounts for this.

Counting a region more than once

Some polar curves retrace points during a full revolution. Make sure your interval traces the desired region exactly once. This is especially important for rose curves and curves with multiple loops.

Practice Problems

Question 1

Find the area enclosed by .

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Question 2

Find the area enclosed by the cardioid .

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Question 3

Find the area of one petal of .

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Question 4 (Symmetry)

Use symmetry to find the area enclosed by .

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