C41 · Unit C9 Parametric, polar and vector calculus
Polar slopes and areas
BC extensionReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 9.7–9.9 · BC
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
For r equals r( theta), convert to x equals r cos theta,y equals r sin theta and differentiate: dy over dx equals (r minutes sin theta +r cos theta)/(r minutes cos theta minus r sin theta) when the denominator is nonzero. Polar area comes from small sectors: A equals one half ∫r squared d theta. For an area between curves on a common angular interval, use one half ∫(r_outer squared minus r_inner squared)d theta, after checking intersection angles, nonnegative radii and tracing.
Worked example
The curve r equals 2cos theta traces a radius-1 circle once for minus pi /2 is less than or equal to theta is less than or equal to pi /2. Its area is one half ∫4cos squared theta d theta equals pi over that interval. Integrating over 0 is less than or equal to theta is less than or equal to 2 pi would trace the same circle twice and double-count area.
Explore the mathematics
Polar slopes and areas
Build a polar area from narrow sectors with radian angles.
Scroll sideways to see the full diagram.
Predict before continuing
A narrow polar slice is a sector, not a rectangle.
Read the complete explanation
- A narrow polar slice is a sector, not a rectangle.
- Sector area is ½r²Δθ, with Δθ in radians.
- Integrate ½r(θ)² over the angular interval.
- Area is ½×4×π/3=2π/3. For shared regions, identify which radius bounds the intersection on each subinterval.
The curve r=2cosθ traces a radius-1 circle once for −π/2≤θ≤π/2. Its area is ½∫4cos²θ dθ=π over that interval. Integrating over 0≤θ≤2π would trace the same circle twice and double-count area.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
A squared radius does not resolve repeated tracing. Curves may also meet at the origin at different parameter values.
Investigate and explain
Trace r equals 2cos theta over two different intervals. Mark the origin and compare accumulated sector area with the geometric circle.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.