For a polar curve described by

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

Problem

For a polar curve described by r=f(θ)r=f(\theta), which of the following integrals represents the exact area of the region traced once as θ\theta varies from α\alpha to β\beta? (a) 12∫αβ(drdθ)2 dθ\displaystyle \frac12\int_{\alpha}^{\beta}\left(\dfrac{dr}{d\theta}\right)^{2}\,d\theta (b) ∫αβr dθ\displaystyle \int_{\alpha}^{\beta} r\,d\theta (c) 12∫αβr2 dθ\displaystyle \frac12\int_{\alpha}^{\beta} r^{2}\,d\theta (d) ∫αβr2 dθ\displaystyle \int_{\alpha}^{\beta} r^{2}\,d\theta

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