A smooth curve is traced once by the vector function for

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

Problem

A smooth curve

C

is traced once by the vector function

\mathbf r(t)=(x(t),y(t))

for

a\le t\le b

. Let

L

denote the arc length of

C

found over the interval

[a,b]

. Which statement is true about the arc length obtained by instead integrating over the interval

[b,a]

? (a) It is

2L

, because the curve is effectively traversed twice. (b) It is

-L

, because reversing the limits changes the sign of the integral. (c) It is equal to

L

; reversing the limits does not change the arc length. (d) It is

0

, since integrating over a negative interval yields zero.