Let be a real number with

MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

Let

a

be a real number with

0<a<1

. Consider the curve

x(t)=t,\; y(t)=a\sin t

0\le t\le 2\pi

. The goal is to establish quantitative bounds on its total length in terms of

a

. (a) Write an expression for the speed of the curve in terms of

t

and

a

. Simplify as far as possible. (b) Use the inequality

|\sin t|\le 1

to obtain an upper bound

U(a)

and a lower bound

L(a)

for the speed that hold for all

t

[0,2\pi]

. State both bounds explicitly. (c) Explain how your bounds from part (b) yield corresponding upper and lower bounds for the total length

S(a)

of the curve on

[0,2\pi]

. Express each bound in simplest closed form. (d) For

a=\tfrac12

, evaluate your bounds numerically and comment on how tightly they constrain

S(\tfrac12)