Let be the curve given by and

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

Problem

Let

C_1

be the curve given by

x(t)=t

and

y(t)=\sin t

, and let

C_2

be the curve given by

x(t)=t

and

y(t)=2\sin t

, each for

0\le t\le \pi

. Which statement best describes the relationship between their arc lengths? (a)

C_1

is longer primarily because its

y

–values oscillate less. (b)

C_1

and

C_2

have the same length over the interval

[0,\pi]

. (c)

C_2

is longer, but by a factor less than

2

. (d)

C_2

is exactly twice as long because its amplitude is doubled.