For the spiral with

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

Problem

For the spiral

r=a\theta

with

a>0

, which expression gives the exact length of the curve from

\theta=0

\theta=2\pi

? (a)

a\left[2\pi\sqrt{4\pi^{2}+1}+\dfrac12\ln\!\left(2\pi+\sqrt{4\pi^{2}+1}\right)\right]

(b)

a\left[\pi\sqrt{4\pi^{2}+1}+\dfrac12\ln\!\left(2\pi+\sqrt{4\pi^{2}+1}\right)\right]

(c)

a\left[\pi\sqrt{4\pi^{2}+1}+\ln\!\left(2\pi+\sqrt{4\pi^{2}+1}\right)\right]

(d)

a\left[\pi\sqrt{4\pi^{2}+1}\right]