9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Use the sector-area idea: for a polar curve r = r(θ), the area of the region between θ = a and θ = b is A = (1/2) ∫[a to b] r(θ)^2 dθ. Think of slicing into tiny sectors of angle dθ with area ≈ 1/2 r^2 dθ. Choose a and b so the integral traces the desired bounded region once (watch for symmetry: use petals/loops or even/odd symmetry to halve work). For things like rose curves, cardioids, limaçons, or inner loops, find θ-intercepts or values where r(θ)=0 to locate limits; if r becomes negative interpret geometrically (it reflects through the pole) and break the integral at zeros. This formula is exactly what the AP CED lists for CHA-5.D (use definite integrals to calculate polar areas). For step-by-step examples and practice, see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q), the Unit 9 overview (https://library.fiveable.me/ap-calculus/unit-9), and thousands of practice problems (https://library.fiveable.me/practice/ap-calculus).

Use the polar sector formula A = (1/2) ∫ r(θ)^2 dθ. For r = 2 cos θ, r ≥ 0 when cos θ ≥ 0, so the curve is traced for θ ∈ [−π/2, π/2] (it’s a circle of radius 1 centered at (1,0)). Compute: A = 1/2 ∫_{−π/2}^{π/2} (2 cos θ)^2 dθ = 1/2 ∫_{−π/2}^{π/2} 4 cos^2 θ dθ = 2 ∫_{−π/2}^{π/2} cos^2 θ dθ = 2 ∫_{−π/2}^{π/2} (1+cos 2θ)/2 dθ = ∫_{−π/2}^{π/2} (1+cos 2θ) dθ = [θ + (1/2) sin 2θ]_{−π/2}^{π/2} = π. So the area enclosed by r = 2 cos θ is π. For more examples and AP-style practice on Topic 9.8, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and extra practice (https://library.fiveable.me/practice/ap-calculus).

Use the polar-area formula A = (1/2) ∫ r(θ)^2 dθ whenever the region is defined naturally by a polar curve r = r(θ) and you can describe the boundary by angles θ = a to θ = b. That formula comes from summing sector areas (area of small sector ≈ 1/2 r^2 Δθ), so it’s the right tool for petals, loops, cardioids, limaçons, etc. (This is exactly CED CHA-5.D: extend area idea to polar coordinates.) Use “regular” (rectangular) definite integrals—A = ∫ (top − bottom) dx or A = ∫ (right − left) dy—when the boundary curves are given as y = f(x) or x = g(y) and are easier to describe with x or y limits. Practical checklist: - Curve given as r(θ) or shape described by θ limits → use A = 1/2 ∫ r^2 dθ. - Curve given as y = f(x) (or vertical slices) → use ∫ (top − bottom) dx. - If region crosses the pole or r(θ) changes sign, split the θ-interval into pieces where r ≥ 0 or each loop/petal is traced exactly. - Use symmetry to shorten work (petals, even/odd sine-cosine). For AP BC Topic 9.8 review and worked examples, see the study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q). For more practice problems, check (https://library.fiveable.me/practice/ap-calculus).

Think of the polar-area integral A = (1/2) ∫ r(θ)^2 dθ as adding up sector slices—the tricky part is choosing θ so the slices cover the region exactly once. How to find limits: - Sketch or graph r(θ). Find θ values where the curve crosses the pole (r = 0) and where it intersects itself or returns to the same point. Those are natural boundaries for a single loop or petal. - For petals/roses: a single petal often lies between consecutive θ where r = 0 (or between symmetry lines). For example many r = a sin(nθ) or cos(nθ) petals are between adjacent zeros. - For cardioids/limaçons: locate θ where r = 0 (inner loop endpoints) or where the curve stops/starts sweeping the bounded region. - Use symmetry: if the graph is symmetric about an axis, integrate one symmetric piece and multiply. - If unsure, graph on [α, α+2π] with a calculator and pick the subinterval that traces the desired region exactly once. This is exactly CHA-5.D territory—practice identifying θ-intercepts and loops on the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and try problems from the unit practice set (https://library.fiveable.me/practice/ap-calculus).

Start with the sector-area idea: a small wedge at angle θ has area ≈ 1/2 r(θ)^2 dθ, so the area of a region traced by r = r(θ) is A = (1/2) ∫ r(θ)^2 dθ. Step-by-step: 1. Sketch or quickly graph r(θ). Identify the region’s boundaries: where the curve meets the pole (r = 0), θ-intercepts, or intersection angles with itself. 2. Choose limits of integration that trace the region exactly once. For a single loop/petal, find θ1 and θ2 with r(θ1)=r(θ2)=0 (or where the loop closes). If symmetry exists (rose, cardioid, limaçon), integrate one symmetric sector and multiply. 3. Set up A = (1/2) ∫_{θ1}^{θ2} [r(θ)]^2 dθ. 4. Evaluate the integral. If the curve has an inner loop, split the integral at the angles where r changes sign or at intersections so each subintegral describes the intended piece of region; use absolute geometry (area is positive). On the AP exam they expect you to justify limits (θ values) and use symmetry when helpful. For examples and practice, see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and Unit 9 overview (https://library.fiveable.me/ap-calculus/unit-9). For extra practice problems try (https://library.fiveable.me/practice/ap-calculus).

Rectangular (Cartesian) area: you slice the region into thin rectangles of width dx (or dy). Each slice’s area ≈ height · width, so A = ∫ (top − bottom) dx (or ∫ (right − left) dy). Polar area: points are (r(θ), θ), so slices are thin sectors, not rectangles. A small sector with angle dθ has area ≈ 1/2 [r(θ)]^2 dθ, so for a region traced by r(θ) you use A = 1/2 ∫ r(θ)^2 dθ with correct θ-limits that cover the region once. Key differences to watch for: choose θ-limits that avoid double-counting loops/petals, use symmetry when possible (multiply a single petal/half by the appropriate factor), and remember r can be negative or produce inner loops—sketch the curve or find θ-intercepts. This aligns with CED CHA-5.D (area in polar coordinates). For a quick review and examples, see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and more unit resources (https://library.fiveable.me/ap-calculus/unit-9). For extra practice, try the AP problems on Fiveable (https://library.fiveable.me/practice/ap-calculus).

You always multiply by 1/2 because polar area comes from the area of a small sector. A sector of radius r and angle Δθ has area ≈ (1/2) r^2 Δθ, so summing (integrating) those sectors gives A = (1/2) ∫ r(θ)^2 dθ (CHA-5.D.1). Important things to check before you apply it: - Use r(θ) = distance from pole for the region swept by θ; pick θ-limits that trace the desired region exactly once (watch petals, loops, cardioids). - If the curve crosses itself or has an inner loop, split the integral at the θ values where r = 0 and add/subtract the appropriate sector areas. - For area between two polar curves r1 and r2 use A = (1/2) ∫ (r1^2 − r2^2) dθ over the θ interval that gives outer/inner consistently. For AP practice and worked examples (including how to pick limits and handle symmetry), see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q), the Unit 9 overview (https://library.fiveable.me/ap-calculus/unit-9), and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Think of polar area like adding up lots of tiny circular sectors. For a small angle Δθ the sector with radius r has area (1/2)·r^2·Δθ (that’s just the standard sector formula). If the curve is r = r(θ), over a tiny slice from θ to θ+Δθ the radius is about r(θ), so the slice’s area ≈ 1/2 · [r(θ)]^2 · Δθ. Summing those slices and taking the limit (Riemann sum → definite integral) gives the polar area formula A = 1/2 ∫_{α}^{β} [r(θ)]^2 dθ. So the r^2 comes from the sector area formula (area scales with the square of the radius). This is exactly what the CED lists for Topic 9.8 (CHA-5.D): use 1/2 ∫ r^2 dθ with correct limits, watch for loops and symmetry. For practice and worked examples, check the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and try problems on the Unit 9 practice page (https://library.fiveable.me/practice/ap-calculus).

Use the polar sector area idea: a small sector has area ≈ 1/2 r^2 dθ, so for two polar curves r1(θ) and r2(θ) the area between them (where router ≥ rinner) on [α, β] is A = (1/2) ∫_{α}^{β} [r_outer(θ)^2 − r_inner(θ)^2] dθ. Steps: 1. Find intersection angles by solving r1(θ) = r2(θ). Those give your limits α, β (or split into pieces if they cross multiple times). 2. For each subinterval determine which r is outer (larger r). If they swap, split the integral at the swap point. 3. Use symmetry (petals, loops, cardioids) when possible to reduce work; the CED expects you to set correct limits and justify them. 4. Evaluate the integral; watch for inner loops where r may be negative—still square r, but be careful with θ-interval choice so the region is traced once. This is the BC Topic 9.8 method (CHA-5.D.1). For worked examples and practice, see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and the AP Calc practice bank (https://library.fiveable.me/practice/ap-calculus).

When a polar curve crosses itself, you must split the region into separate sectors (loops) and integrate each one with A = (1/2) ∫ r(θ)^2 dθ using the correct θ-limits—don’t try to use one big interval that double-counts area. How to do that in practice: - Find intersection θ-values by solving r(θ1)=r(θ2) with θ1 ≠ θ2 (and include where r(θ)=0, the pole). Those θs mark boundaries of distinct loops or petals. - Sketch or use technology to see which θ-interval traces each bounded region. Each loop’s area = (1/2) ∫_{α}^{β} r(θ)^2 dθ where [α,β] is the interval that traces that loop exactly once. - Use symmetry (e.g., rose, cardioid, limaçon) to reduce work: compute one petal/loop and multiply. - Watch inner loops: if r becomes negative, use the same formula but ensure your α,β correspond to the correct tracing (r^2 handles sign, but limits must match one traversal). This matches the AP CED skill CHA-5.D (use 1/2 ∫ r^2 dθ with correct limits). For worked examples and extra practice on splitting intervals and loops, see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and more practice problems at (https://library.fiveable.me/practice/ap-calculus).

Use the sector formula A = (1/2) ∫ r(θ)^2 dθ—that’s the AP CED formula (CHA-5.D.1). On your graphing calculator: 1. Decide limits θ = a to b (find intersections/pole crossings). If the curve loops or r changes sign, split the interval at r(θ)=0 or at intersection angles so each integral covers one simple region. Use your calculator’s polar graph to find those θ-values (trace/zero or intersection). 2. In radian mode (match the problem), go to the numeric integrator: enter (1/2)*(r(θ))^2 with r(θ) written in terms of θ (e.g., (1/2)*(2+4*sin(theta))^2). Use your calc’s ∫ function and evaluate from a to b. 3. Use symmetry to reduce work (petals/loops often repeat). If you get a negative integrand because r<0, split at zeros so r^2 stays valid. On the AP exam, a graphing calculator is required for Part B (where numeric work like this may appear). For step-by-step examples and practice, see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and try problems at (https://library.fiveable.me/practice/ap-calculus).

You’re probably making one of two common mistakes. 1) Using the wrong integrand. The area formula is A = (1/2) ∫ r(θ)^2 dθ. If you accidentally use r or forget the square, the sign of r can give a negative result. r^2 is always ≥ 0, so the integrand itself shouldn’t be negative. 2) Wrong limits (or not splitting where r = 0). Polar curves can loop and r can change sign. A single integral over an interval where the curve traces different regions (or traces a region more than once) can give a “negative-looking” answer if you: - choose θ-limits that don’t correspond to the single bounded region, or - interpret parts where r < 0 incorrectly (remember r < 0 reflects the point across the origin by π). How to fix it: - Sketch the curve or use a graphing calculator and find θ where r(θ)=0 to locate loop endpoints. - Integrate (1/2)r^2 over the θ-interval that traces exactly the region once. If r changes sign inside, that’s okay—r^2 stays positive—but you must pick the correct bounds. - Use symmetry (petals/loops) to reduce work. For step-by-step examples and practice, check the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and Unit 9 overview (https://library.fiveable.me/ap-calculus/unit-9). If you want, tell me the specific r(θ) and limits you used and I’ll check them.

Pick bounds so your integral covers the region exactly once. Quick checklist you can use every time: 1. Identify where the curve starts/ends one trace—find θ where r(θ) repeats its values. For many trig polar formulas use their period (e.g., r = cos(nθ): if n is odd trace from 0 to π, if n even trace 0 to π/2). 2. Solve r(θ) = 0 and other θ-intercepts to locate boundaries of loops/petals or an inner loop (these zeros often split the region). 3. Use symmetry: if the curve is symmetric about an axis or the pole, integrate one symmetric sector and multiply (saves work and reduces bounds). 4. Make sure the chosen θ-interval traces the bounded region exactly once (no overlaps). Then apply A = 1/2 ∫_{α}^{β} [r(θ)]^2 dθ. 5. If unsure, sketch (or use graphing tech on calculator-allowed parts of the AP BC exam) to confirm which θ give the intended loop. For more worked examples and AP-aligned tips for Topic 9.8, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q). For extra practice, use the unit practice bank (https://library.fiveable.me/practice/ap-calculus).

Use the polar-sector area idea: for an interval of θ where one curve lies outside the other, the area between r1(θ) and r2(θ) is A = 1/2 ∫[θ=a to b] (r_outer(θ)^2 − r_inner(θ)^2) dθ. Setup steps: 1. Solve r1(θ) = r2(θ) to get intersection angles; those give candidate limits a and b. 2. On each candidate interval check which r is larger (r_outer)—remember r can be negative in polar form, so interpret points/angles carefully. 3. Integrate 1/2 (r_outer^2 − r_inner^2) over each interval where the ordering is consistent; add results if region spans multiple intervals. 4. Use symmetry (petals, loops, cardioids) to reduce work when possible. This matches the AP CED skill CHA-5.D (area in polar coords). For walk-throughs and examples, see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q) and plenty of practice problems at (https://library.fiveable.me/practice/ap-calculus).

Use the polar-sector idea: a small sector with radius r(θ) and angle dθ has area ≈ 1/2 r(θ)^2 dθ. So for a region traced by a single polar curve r = r(θ) between θ = a and θ = b the area is A = (1/2) ∫[a to b] [r(θ)]^2 dθ. When to use it - Use this whenever the boundary is a single polar curve given as r(θ) and you want the area of the region it sweeps for θ from a to b (petals, loops, cardioids, limaçons, etc.). - Pick a and b so the integral traces the region exactly once (watch inner loops where r can be 0 or negative). Use symmetry (e.g., petals of roses) to reduce work. - If the curve crosses the pole, find θ-intercepts (solve r(θ)=0) to set correct limits. This is a BC-only topic in the CED (CHA-5.D). For a focused review and examples, see the Topic 9.8 study guide (https://library.fiveable.me/ap-calculus/unit-9/find-area-polar-region-or-area-bounded-by-single-polar-curve/study-guide/XhdT4ohGZFpzjdRT2l2q). For broader unit review and tons of practice, check the unit page (https://library.fiveable.me/ap-calculus/unit-9) and the practice bank (https://library.fiveable.me/practice/ap-calculus).