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Unit 9.9 – Area Bounded by Two Polar Curves BC ONLY

AP® Calculus BC | Finding Areas Between Polar Curves

Why This Matters: Finding the area between two polar curves is THE ultimate challenge in polar calculus! You must find intersection points, determine which curve is outer/inner (and this can change!), and set up proper integrals. This topic combines everything you know about polar coordinates and appears on virtually every BC exam as a challenging free-response question!

📐 The Area Between Two Polar Curves

Area Between Curves Formula

THE FUNDAMENTAL FORMULA:

For curves \(r = f(\theta)\) and \(r = g(\theta)\) from \(\theta = \alpha\) to \(\theta = \beta\):

\[ A = \frac{1}{2}\int_\alpha^\beta \left([r_{\text{outer}}]^2 - [r_{\text{inner}}]^2\right) \, d\theta \]
\[ A = \frac{1}{2}\int_\alpha^\beta \left([f(\theta)]^2 - [g(\theta)]^2\right) \, d\theta \]

⚠️ CRITICAL: This is NOT \(\frac{1}{2}\int(r_{\text{outer}} - r_{\text{inner}})^2\,d\theta\)! You must square EACH function separately, then subtract!

📋 Complete Step-by-Step Process

The 6-Step Method

Follow This Process Every Time:

  1. Sketch both curves: Visualize the region (critical!)
  2. Find intersection points: Solve \(f(\theta) = g(\theta)\)
  3. Determine bounds: Use intersection angles as limits
  4. Identify outer and inner curves: Check which is farther from origin
  5. Set up integral: \(\frac{1}{2}\int(r_{\text{outer}}^2 - r_{\text{inner}}^2)\,d\theta\)
  6. Evaluate: Use power-reduction if needed

🔍 Finding Intersection Points

INTERSECTION METHODS

Three Ways Curves Can Intersect:

1. Simultaneous Intersection:

Both curves at same point with same \(\theta\):

\[ f(\theta) = g(\theta) \]

Solve algebraically for \(\theta\)

2. At the Pole (Origin):

Find where each curve passes through \(r = 0\):

  • For curve 1: \(f(\theta) = 0\)
  • For curve 2: \(g(\theta) = 0\)

They may pass through origin at different angles!

3. Different Representations:

Point \((r, \theta)\) is same as \((-r, \theta + \pi)\)

Check if curves meet with one positive \(r\) and one negative \(r\)

📝 Important: Always check for intersection at the origin separately! Many problems have curves passing through the pole at different angles.

🎯 Determining Which Curve is Outer

Critical Decision:

Method 1: Test a Value

Pick a \(\theta\) value between intersections and evaluate both \(r\) values:

  • Larger \(|r|\) value → outer curve
  • Smaller \(|r|\) value → inner curve
Method 2: Sketch

Graph both curves and visually identify which is farther from origin in the region of interest

⚠️ WARNING: The outer and inner curves can SWITCH! You may need multiple integrals for different regions where different curves are outer.

📖 Comprehensive Worked Examples

Example 1: Inside One Curve, Outside Another

Problem: Find the area inside \(r = 2 + 2\sin\theta\) and outside \(r = 2\).

Solution:

Step 1: Find intersections

\[ 2 + 2\sin\theta = 2 \]
\[ \sin\theta = 0 \quad \Rightarrow \quad \theta = 0, \pi, 2\pi \]

By symmetry, we can integrate from 0 to \(\pi\) and double.

Step 2: Determine outer/inner

Test \(\theta = \pi/2\):

\(r_1 = 2 + 2\sin(\pi/2) = 4\) (outer)

\(r_2 = 2\) (inner)

Step 3: Set up integral

\[ A = \frac{1}{2}\int_0^\pi \left[(2+2\sin\theta)^2 - 2^2\right] \, d\theta \]
\[ = \frac{1}{2}\int_0^\pi \left[4 + 8\sin\theta + 4\sin^2\theta - 4\right] \, d\theta \]
\[ = \frac{1}{2}\int_0^\pi \left(8\sin\theta + 4\sin^2\theta\right) \, d\theta \]

Step 4: Use \(\sin^2\theta = \frac{1-\cos(2\theta)}{2}\)

\[ A = \frac{1}{2}\int_0^\pi \left(8\sin\theta + 2 - 2\cos(2\theta)\right) \, d\theta \]
\[ = \frac{1}{2}\left[-8\cos\theta + 2\theta - \sin(2\theta)\right]_0^\pi \]
\[ = \frac{1}{2}[(8 + 2\pi) - (-8)] = \frac{1}{2}(16 + 2\pi) = 8 + \pi \]

Example 2: Two Curves with Multiple Intersections

Problem: Find the area inside both \(r = 2\cos\theta\) and \(r = 2\sin\theta\).

Find intersections:

\[ 2\cos\theta = 2\sin\theta \quad \Rightarrow \quad \tan\theta = 1 \quad \Rightarrow \quad \theta = \frac{\pi}{4} \]

Also both pass through origin at different angles:

  • \(r = 2\cos\theta = 0\) at \(\theta = \pi/2\)
  • \(r = 2\sin\theta = 0\) at \(\theta = 0\)

Setup (using symmetry and proper bounds):

The region "inside both" is the lens-shaped overlap.

From \(\theta = 0\) to \(\theta = \pi/4\): \(r = 2\sin\theta\) is outer

\[ A = 2 \cdot \frac{1}{2}\int_0^{\pi/4} (2\sin\theta)^2 \, d\theta = \int_0^{\pi/4} 4\sin^2\theta \, d\theta \]

(Factor of 2 for the other half by symmetry)

Example 3: Rose Curves Intersection

Problem: Find the area inside \(r = 2\cos(2\theta)\) and outside \(r = 1\).

Find intersections:

\[ 2\cos(2\theta) = 1 \quad \Rightarrow \quad \cos(2\theta) = \frac{1}{2} \]
\[ 2\theta = \pm\frac{\pi}{3}, \pm\frac{5\pi}{3} \quad \Rightarrow \quad \theta = \pm\frac{\pi}{6}, \pm\frac{5\pi}{6} \]

Setup using symmetry:

Rose curve has 4 petals. Due to symmetry, find area for one region and multiply by 4:

\[ A = 4 \cdot \frac{1}{2}\int_{-\pi/6}^{\pi/6} \left[(2\cos(2\theta))^2 - 1^2\right] \, d\theta \]

Example 4: Common Tangent at Origin

Problem: Find area inside \(r = 1 + \cos\theta\) and outside \(r = 3\cos\theta\).

Analysis:

Both curves pass through origin:

  • \(1 + \cos\theta = 0\) at \(\theta = \pi\)
  • \(3\cos\theta = 0\) at \(\theta = \pi/2\)

Find other intersections algebraically.

🎨 Common Area Scenarios

Typical Problem Types
Scenario Setup Key Point
Inside A, Outside B \(\frac{1}{2}\int(r_A^2 - r_B^2)\,d\theta\) A is outer, B is inner
Inside Both Find overlap region Use intersection points carefully
Between Two May need multiple integrals Outer/inner can switch!
Rose Petals Use symmetry Multiply single petal area

💡 Essential Tips & Strategies

✅ Success Strategies:

  • ALWAYS sketch first: Non-negotiable for these problems!
  • Find ALL intersections: Including at the origin
  • Test which is outer: Use a specific \(\theta\) value
  • Check if outer/inner switches: May need split integrals
  • Use symmetry: Reduces calculation errors
  • Don't forget \(\frac{1}{2}\): Still applies!
  • Square EACH function: Then subtract
  • Expand carefully: \((a + b)^2 \neq a^2 + b^2\)

🔥 Problem-Solving Checklist:

  1. Sketch both curves
  2. Solve \(f(\theta) = g(\theta)\) for intersections
  3. Check for intersection at origin separately
  4. Identify the region clearly
  5. Test to determine outer/inner
  6. Write integral with correct bounds
  7. Verify setup makes sense with sketch
  8. Integrate and simplify

❌ Common Mistakes to Avoid

  • Mistake 1: Not sketching the curves (biggest error!)
  • Mistake 2: Missing intersection at the origin
  • Mistake 3: Using \(\frac{1}{2}\int(r_1 - r_2)^2\) instead of \(\frac{1}{2}\int(r_1^2 - r_2^2)\)
  • Mistake 4: Not checking which curve is outer
  • Mistake 5: Forgetting outer/inner can switch
  • Mistake 6: Wrong bounds from incorrect intersection points
  • Mistake 7: Forgetting the \(\frac{1}{2}\) factor
  • Mistake 8: Not using symmetry when available
  • Mistake 9: Expanding \((a+b\cos\theta)^2\) incorrectly
  • Mistake 10: Calculator in degree mode

📝 Practice Problems

Find the area:

  1. Inside \(r = 3 + 3\sin\theta\) and outside \(r = 3\)
  2. Inside both \(r = 4\cos\theta\) and \(r = 4\sin\theta\)
  3. Inside \(r = 2\) and outside \(r = 2\cos\theta\)
  4. Common interior of \(r = 1\) and \(r = 2\sin\theta\)

Answers:

  1. \(9\pi + \frac{27}{2}\)
  2. \(4\pi - 8\)
  3. \(2\pi - \frac{3\sqrt{3}}{2}\)
  4. \(\frac{\pi}{3} + \frac{\sqrt{3}}{2}\)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

  • Show intersection work: Set equations equal, solve for \(\theta\)
  • State which is outer/inner: "For \(\theta \in [a,b]\), \(r_1 > r_2\)"
  • Write complete formula: \(A = \frac{1}{2}\int_a^b(r_{\text{outer}}^2 - r_{\text{inner}}^2)\,d\theta\)
  • Show squaring: Write out \([f(\theta)]^2\) and \([g(\theta)]^2\)
  • Show expansion: When squaring binomials
  • Use symmetry justification: If applicable
  • Show integration: Or clearly state using calculator
  • Simplify final answer: Exact form with \(\pi\)

💯 Exam Strategy:

  1. READ CAREFULLY: "Inside A and outside B" vs "Inside both"
  2. Sketch both curves immediately
  3. Find all intersection points
  4. Shade the desired region on sketch
  5. Determine bounds from intersections
  6. Test to find outer/inner curves
  7. Set up integral correctly
  8. Verify setup matches your sketch
  9. Integrate carefully
  10. Check answer reasonableness

⚡ Quick Reference Guide

AREA BETWEEN POLAR CURVES ESSENTIALS

Main Formula:

\[ A = \frac{1}{2}\int_\alpha^\beta (r_{\text{outer}}^2 - r_{\text{inner}}^2) \, d\theta \]

Finding Intersections:

  • Simultaneous: \(f(\theta) = g(\theta)\)
  • At origin: \(f(\theta) = 0\) and \(g(\theta) = 0\)
  • Check different representations

Determining Outer:

Test a \(\theta\) value between intersections

Larger \(|r|\) = outer curve

Remember:

  • SKETCH FIRST!
  • Find ALL intersections
  • Square each, then subtract
  • Outer/inner can switch!

Master Area Between Polar Curves! The formula: \(A = \frac{1}{2}\int_\alpha^\beta(r_{\text{outer}}^2 - r_{\text{inner}}^2)\,d\theta\) where you square EACH function then subtract (NOT square the difference!). Critical steps: (1) SKETCH both curves—absolutely essential! (2) Find ALL intersections by solving \(f(\theta) = g(\theta)\) AND checking origin separately. (3) Test a \(\theta\) value to determine which curve is outer (larger \(|r|\)). (4) Watch for outer/inner switching—may need multiple integrals! (5) Use symmetry when possible. Common language: "inside A and outside B" means A is outer, B is inner. "Inside both" means find overlap region. "Between" may need careful analysis. Most common errors: not sketching, missing origin intersection, using \((r_1-r_2)^2\) instead of \(r_1^2-r_2^2\), wrong outer/inner identification. This is THE hardest polar topic—appears as challenging FR every year! Practice extensively until intersection-finding and setup become automatic! 🎯✨