Let be a differentiable function on the closed interval with continuous on

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

Problem

Let

f

be a differentiable function on the closed interval

[a,b]

with

f'(x)

continuous on

[a,b]

. A student partitions

[a,b]

into

n

subintervals of equal width and joins successive points

(x_i,f(x_i))

with straight segments to form a polygonal path along the graph of

y=f(x)

. The length of this polygonal path is denoted

L_n

. Assume

L_n

approaches a finite limit as

n\to\infty

. (a) Describe, in words and with appropriate limiting notation, how

L_n

is used to motivate an exact expression for the length

L

of the graph of

y=f(x)

from

x=a

x=b

. Explain the role played by the Mean Value Theorem in the derivation. (b) Use your reasoning from part (a) to write a definite integral that equals

L

. Clearly state the integrand and the limits of integration. (c) Let

f(x)=3x+1

[1,5]

. Evaluate the integral you obtained in part (b) to find the exact length of the graph of

f

on this interval, and explain why the result agrees with elementary geometry.