Practice Problem Set: Polar Curves and Area

What This Problem Set Covers

Polar-coordinate problems on the AP Calculus BC exam require you to move comfortably between an equation in , geometric interpretation, and calculus. The hardest part is often not the integration itself. It is deciding what interval to use, recognizing symmetry, and identifying exactly which region the problem is asking about.

This practice set brings those skills together. You should be comfortable with the polar area formula, the derivative of a polar curve, and the idea that a single polar equation can trace a curve more than once.

Essential Formulas

Keep these formulas available while working the problems. On an AP free-response question, the important skill is knowing when each formula applies.

The area formula is especially important. A polar area integral contains one-half the square of the radius, not simply the radius. The bounds must also correspond to the portion of the curve being counted.

A Reliable Strategy

Sketch or visualize the curve. Determine whether it is a circle, cardioid, rose, limacon, or another recognizable polar curve.
Look for symmetry. Symmetry can reduce the interval you need to analyze.
Find important angles. Solve for where , where two curves intersect, or where the radius changes sign.
Determine the correct bounds. Do not automatically use . The correct interval depends on what region is being traced.
Set up the integral before calculating. A correct setup is often the most important part of the problem.
Check whether the curve is traced once. Some polar equations trace their entire curve over an interval shorter than , while others require a longer interval.

Worked Examples

Example 1: Area Inside a Polar Curve

Find the area enclosed by the cardioid .

Step 1: The curve is symmetric about the polar axis, so we can calculate the upper half and double it.

Step 2: Using the polar area formula:

Step 3: Therefore:

Step 4: Expand the square:

Result:

Example 2: Area of One Petal of a Rose

Find the area of one petal of .

Step 1: For , the curve has five petals. One petal is traced between consecutive zeros of the radius.

Step 2: Set the radius equal to zero:

Step 3: One convenient interval for a single petal is .

Step 4: Set up the area:

Result:

Example 3: Area Between Two Polar Curves

Find the area inside but outside .

Step 1: Find the intersection angles:

Step 2: Determine which curve has the larger radius on the interval. Between the intersection angles, is the outer curve.

Step 3: Subtract the inner area from the outer area:

Result:

Example 4: Slope of a Polar Curve

For , find the slope of the curve at .

Step 1: Use the polar derivative formula:

Step 2: Differentiate the radius:

Step 3: Substitute the angle:

Result:

Common Mistakes

Forgetting the one-half in the area formula

The polar area formula is . Do not use . The square and the one-half are both essential.

Using the wrong interval

A polar curve may be traced more than once, or a requested region may correspond to only part of a complete cycle. Always determine where the region begins and ends before integrating.

Subtracting the wrong curves

For area between two polar curves, identify which curve is farther from the origin on the interval. The correct setup is , with the larger radius squared first.

Ignoring negative values of r

A negative polar radius does not automatically mean the curve has stopped. The point represents the point obtained by moving in the opposite direction from the angle. This is one reason sketches and symmetry checks are so useful.

Using the Cartesian derivative formula

For a polar curve, do not use directly. The correct formula is .

Practice Problems

Question 1 — Polar Area

Find the area enclosed by the circle .

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Question 2 — Area of a Rose Petal

Find the area of one petal of the rose .

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Question 3 — Symmetry

Determine the symmetry of the polar curve and explain how that symmetry can be used when finding its total area.

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Question 4 — Polar Derivative

Find the slope of the curve at .

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Question 5 — Horizontal Tangent

Find the values of for which the polar curve has a horizontal tangent.

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Question 6 — Intersections

Find the intersection points of and for .

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Question 7 — Area Between Polar Curves

Find the area inside but outside .

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Question 8 — Region Inside a Cardioid

Find the area enclosed by .

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Question 9 — Inner Loop

Find the area of the inner loop of .

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Question 10 — AP-Style Setup

A particle moves along the polar curve . Set up, but do not evaluate, an integral representing the area of the region traced as .

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How to Check Your Work

Before accepting a polar-area answer, perform three quick checks.

Check the bounds. Ask whether the interval traces exactly the region requested.
Check the integrand. For area, the radius should normally appear squared.
Check the geometry. Your answer should be consistent with the approximate size of the region. A symmetry argument can often provide a useful sanity check.

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