For , let and . Consider the region between the graphs of and from to , where…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

For

x\ge 0

, let

f(x)=x^{2}

and

g(x)=2x

. Consider the region between the graphs of

f

and

g

from

x=0

x=k

, where

k>0

. (a) Express

\displaystyle A_f(k)=\int_{0}^{k} f(x)\,dx

and

\displaystyle A_g(k)=\int_{0}^{k} g(x)\,dx

as explicit formulas in terms of

k

. Simplify each result. (b) Find the positive value of

k

for which

A_f(k)=A_g(k)

. Show all steps leading to your solution. (c) For the value of

k

found in part (b), describe the geometric relationship between the two regions bounded by the graphs of

f

and

g

and the

x

-axis on

[0,k]

. Explain your reasoning.