Finding the Area of the Region Bounded by Two Polar Curves

Identify the intended common-ray region, changing boundaries, and correct squared-radius differences.

College Board Topic 9.9 · CED p. 174 (PDF p. 179) · CED effective Fall 2020; project course outline 2026–27.

On a common ray with $0 \leq r_{i n n e r} (θ) \leq r_{o u t e r} (θ)$ , the thin area between two boundaries is $\frac{1}{2} ({r_{o u t e r}}^{2} - {r_{i n n e r}}^{2}) d θ$ . Integrate over the angles that sweep the named region. You subtract squared radii, not square the difference of the radii.

Find candidate boundary intersections by equating radii at the same angle, then inspect the actual points and the pole. Polar curves can meet using different angle/signed-radius descriptions, and both can pass through the pole at different angles. A graph or coordinate conversion helps detect these cases. Break the integral where the outer boundary changes.

For overlap (inside both), the radial upper bound is the smaller nonnegative boundary on each ray. For inside one and outside another, the larger curve must be the outer one over the chosen interval. If the radial region does not begin at the pole or signed radii are involved, rewrite as radial pieces rather than trusting a global max/min rule.

A calculator can approximate an intersection and integral. Record a defining equation and interval, verify which boundary is outside using a test angle, and then integrate. Rounded intersection angles can propagate an error; retain full stored precision until the final decimal.

Let's work through it

Find the overlap of $r = 1$ and $r = 2 \cos θ$ . The second disk has nonnegative radius for $- π / 2 \leq θ \leq π / 2$ . Why: Those angles cover the right-hand disk once.
Solve $1 = 2 \cos θ$ to get $θ = \pm π / 3$ . Near θ=0 the unit circle is the smaller bound; near π/2 the other circle is smaller. Why: Overlap uses the smaller radial upper boundary, which changes at intersections.
By symmetry, $A = {\int_{0}}^{π / 3} 1 d θ + {\int_{π / 3}}^{π / 2} 4 \cos^{2} θ d θ = 2 π / 3 - \sqrt{3} / 2$ . Why: The original 1/2 factor cancels the symmetry factor 2; integrate the two different squared upper bounds.

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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.

Question 1 of 20 self-assessed

Question 1

Between radii 5 and 3 on the same ray, which sector factor belongs in the integrand?

Your answer and reasoning

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Question 1: prompt and worked reasoning

Between radii 5 and 3 on the same ray, which sector factor belongs in the integrand?

(25 - 9) / 2 = 8

Subtract r² values before multiplying by dθ.

Question 2: prompt and worked reasoning

For a region inside both nonnegative curves, do you use the larger or smaller radial upper bound?

The smaller bound, piecewise if necessary.

A point must satisfy both radial inequalities.

See every recall question with its answer

These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.

Retrieval 1

Between radii 5 and 3 on the same ray, which sector factor belongs in the integrand?

Worked answer and reason

(25 - 9) / 2 = 8

Subtract r² values before multiplying by dθ.

Retrieval 2

For a region inside both nonnegative curves, do you use the larger or smaller radial upper bound?

Worked answer and reason

The smaller bound, piecewise if necessary.

A point must satisfy both radial inequalities.

Catch the tempting mistake

Area between curves is 1/2∫(outer−inner)²dθ.: A sector difference is 1/2(outer²−inner²)dθ; the cross term in a squared difference would give the wrong area.

Calculator workspace and limits

For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.

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Put it to work

Unit 9 problem 27 · No calculator · analytic
Unit 9 problem 28 · No calculator · analytic
Unit 9 problem 29 · No calculator · multiple
Unit 9 problem 30 · Graphing calculator required · analytic

For the teacher

Use a two-color sketch or a verbal ray-by-ray description. Assess intersection reasoning, piecewise bounds and the 1/2 factor independently.

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Let's work through it

Try it from memory

Try it before you peek.

Question 1

Retrieval 1

Retrieval 2

Catch the tempting mistake

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