A region in the plane is formed by two non-overlapping sectors: one bounded by the…

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A region in the plane is formed by two non-overlapping sectors: one bounded by the curve r=f(θ)r=f(\theta) on [a,b][a,b] and another bounded by r=g(θ)r=g(\theta) on [b,c][b,c], where ff and gg are non-negative and continuous. Which of the following statements best explains why the total area of the region equals 12∫ab[f(θ)]2dθ+12∫bc[g(θ)]2dθ\displaystyle \frac12\int_{a}^{b} [f(\theta)]^{2} d\theta + \frac12\int_{b}^{c} [g(\theta)]^{2} d\theta? (a) The factor 12\tfrac12 in each integral guarantees no double counting occurs, so the two integrals may be added even if the sectors overlap. (b) Because the two curves never attain the same radius value, adding their integrals cannot over-count any region. (c) Each integral accumulates area only over the angles where its curve is traced, and because the intervals are disjoint the areas add without overlap. (d) Since f(θ)f(\theta) and g(θ)g(\theta) are always non-negative, the sum of their squares automatically produces the total area without further conditions.

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