A differentiable function is strictly increasing on

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

Problem

A differentiable function

y=h(x)

is strictly increasing on

[a,b]

. Its inverse is written

x=g(y)

and is differentiable on

[c,d]

, where

c=h(a)

and

d=h(b)

. A student decides to set up the integral for the length of the graph of

h

from

(a,h(a))

(b,h(b))

by treating

x

as a function of

y

rather than the usual approach of treating

y

as a function of

x

. (a) Express the limits of integration in terms of

y

for the student’s approach and explain how these limits relate to the original interval

[a,b]

. Include a clear description of the geometric meaning of each endpoint. (b) Describe precisely how the differential element that appears in the integrand changes when one switches from treating

y

as the dependent variable to treating

x

as the dependent variable. In particular, identify which derivative replaces

\frac{dy}{dx}

and explain why. (c) Provide a reasoned argument—grounded in the meaning of differentials in the Cartesian plane—that justifies why either orientation (treating

y

as a function of

x

or vice-versa) yields the same numerical value for the length of the curve.

Related theory

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