A student claims that the area between two polar curves and on can be found by integrating

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A student claims that the area between two polar curves r=f(θ)r=f(\theta) and r=g(θ)r=g(\theta) on α≤θ≤β\alpha\le \theta \le \beta can be found by integrating ∫αβ(f(θ)−g(θ)) dθ\int_{\alpha}^{\beta}\bigl(f(\theta)-g(\theta)\bigr)\,d\theta. Another student disagrees and asserts that squaring the radii is essential. Address the dispute by responding to the parts below. (1) For a single polar curve r=h(θ)r=h(\theta), explain—using geometric reasoning—why the differential element of area for a small angle dθd\theta is proportional to h(θ)2 dθh(\theta)^2\,d\theta rather than to h(θ) dθh(\theta)\,d\theta. (2) Using your reasoning from part (a), derive an expression (in integral form) that correctly represents the area between two polar curves on α≤θ≤β\alpha\le \theta \le \beta. Do not simplify the expression. (3) Provide a specific example of two curves on a closed interval of θ\theta where integrating the difference of the radii (without squaring) gives a numerical result different from the correct area. Compute both the incorrect and correct results, and explain the discrepancy.

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