For a smooth curve defined in polar coordinates by

MATHEMATICS · Single correct · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

For a smooth curve defined in polar coordinates by r=f(θ)r=f(\theta), which of the following expressions represents the differential element of arc length dsds in terms of θ\theta? (a) ds=r dθds=r\,d\theta (b) ds=r2+(drdθ)2 dθds=\sqrt{r^{2}+\left(\dfrac{dr}{d\theta}\right)^{2}}\,d\theta (c) ds=r1+(drdθ)2 dθds=r\sqrt{1+\left(\dfrac{dr}{d\theta}\right)^{2}}\,d\theta (d) ds=r2+drdθ dθds=\sqrt{r^{2}+\dfrac{dr}{d\theta}}\,d\theta

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