Two differentiable functions and are defined on with and

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

Two differentiable functions

r

and

s

are defined on

[a,b]

with

r(a)=s(a)

and

r(b)=s(b)

. Suppose

r

and

s

intersect at the additional points

x_{1},x_{2},\dots ,x_{k}

, where

a<x_{1}<x_{2}<\dots <x_{k}<b

, and that

r(x)\ne s(x)

for every

x

not equal to one of these points. (a) Explain why the quantity

\displaystyle\int_{a}^{b}|r(x)-s(x)|\,dx

always equals the sum of

k+1

definite integrals of the form

\int_{x_{j}}^{x_{j+1}}\bigl|r(x)-s(x)\bigr|\,dx

, where

x_{0}=a

and

x_{k+1}=b

. (b) Describe a procedure for calculating the exact area between the graphs of

r

and

s

[a,b]

without using the absolute value notation in the integrand. Be sure to state how one decides which integrand to subtract from which on each sub-interval. (c) Give one example of a pair of elementary functions that intersect at more than two points, and briefly outline how your procedure from part (b) would apply to your example.

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