AP Calculus BC — Unit 9.8
Find the area of a polar region or the area bounded by a single polar curve
Key Concepts Packet + Worked Examples + Interactive MCQ Practice (36 Questions)
Table of Contents
Unit 9.8 Key Concepts Packet
1) Polar Area Formula (Single Curve)
Must-know formula
Area of a region traced once by \(r(\theta)\) from \(\theta=\alpha\) to \(\theta=\beta\):
\( \displaystyle A=\frac12\int_{\alpha}^{\beta}\big(r(\theta)\big)^2\,d\theta \).
- The integrand is \(r^2\), so it is nonnegative even if \(r\) is negative.
- The most common error is forgetting the \(\frac12\).
2) Choosing Correct Bounds (Tracing Once)
Key idea
The bounds \([\alpha,\beta]\) must trace the region exactly once.
- Some curves repeat every \(2\pi\), others every \(\pi\) or smaller depending on symmetry.
- Rose curves \(r=a\cos(n\theta)\) or \(r=a\sin(n\theta)\) often require careful “one petal” bounds.
- Always confirm the curve starts/ends at the pole or repeats a key point before using symmetry shortcuts.
3) Handling Negative Radius and Symmetry
Negative radius identity
\((r,\theta)\equiv(-r,\theta+\pi)\).
- Negative \(r\) can “flip” the point to the opposite direction; this affects how the curve is traced.
- Even though \(r^2\) is positive, the bounds still matter (to avoid double-counting).
- Use symmetry when valid: compute area for a symmetric part, then multiply.
4) Area Bounded by a Single Curve (Typical AP Setups)
Most common AP situations
- Area inside one loop (e.g., cardioid loop, limacon inner loop): choose bounds where the loop is traced once.
- Area of one petal (rose curve): find consecutive zeros of \(r(\theta)\) for one petal.
- Total area: compute one part and multiply if symmetry applies, or integrate over a full tracing interval.
5) Calculator Expectations + Common AP Traps
- AP may require numerical evaluation of \( \frac12\int r^2 d\theta \) on the calculator.
- Trap: Using \( \int r\,d\theta \) or forgetting to square.
- Trap: Integrating over an interval that traces the region twice.
- Important: If you ever see raw backslashes instead of formatted math, it means MathJax did not typeset that section—this page forces typesetting on load and when solutions open.
Worked Examples + Notes
Example 1 — Area inside a cardioid
Find the area inside \(r(\theta)=1+\cos\theta\).
Step 1: This cardioid is traced once on \([0,2\pi]\).
Step 2: Use \(A=\frac12\int_0^{2\pi}(1+\cos\theta)^2\,d\theta\).
Step 3: Expand: \((1+\cos\theta)^2=1+2\cos\theta+\cos^2\theta\).
Step 4: Use \(\cos^2\theta=\frac{1+\cos2\theta}{2}\) and integrate over \([0,2\pi]\).
Conclusion: The setup is the key AP skill: \(A=\frac12\int_0^{2\pi}r^2\,d\theta\) with correct bounds.
Example 2 — Area of one petal (rose curve)
Find the area of one petal of \(r(\theta)=2\sin(2\theta)\).
Step 1: One petal occurs between consecutive zeros of \(r\).
Step 2: Solve \(2\sin(2\theta)=0\Rightarrow \sin(2\theta)=0\Rightarrow 2\theta=0,\pi\).
Step 3: So one petal is traced on \(\theta\in[0,\frac{\pi}{2}]\).
Step 4: Area: \(A=\frac12\int_0^{\pi/2}\big(2\sin(2\theta)\big)^2\,d\theta\).
Conclusion: Correct bounds come from “one loop/petal” tracing, not automatically \([0,2\pi]\).
Example 3 — Using symmetry to simplify
For \(r(\theta)=1+\sin\theta\), explain how symmetry can help compute area.
Step 1: The curve is symmetric about the \(y\)-axis because \(r(\pi-\theta)=1+\sin(\pi-\theta)=1+\sin\theta\).
Step 2: Compute area on \([-\frac{\pi}{2},\frac{\pi}{2}]\) and double, or integrate over \([0,2\pi]\) directly.
Conclusion: Symmetry can reduce computation, but bounds must still trace each region exactly once.
Example 4 — Common mistake check
A student computes \( \int_0^{2\pi}(1+\cos\theta)\,d\theta \) for area inside \(r=1+\cos\theta\). What is the issue?
Step 1: Polar area requires \( \frac12\int r^2\,d\theta \), not \( \int r\,d\theta \).
Step 2: The radius must be squared because area of a thin sector is \( \frac12 r^2\,d\theta \).
Conclusion: Correct setup is \( \frac12\int_0^{2\pi}(1+\cos\theta)^2\,d\theta \).
Note: This page forces MathJax typesetting on load and when solution panels open.
Unit 9.8 Multiple-Choice Practice (36 Questions)
Answer Key