Let and be continuous on the symmetric interval (where ) such that the difference is…

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

Problem

Let

f

and

g

be continuous on the symmetric interval

[-a,a]

(where

a>0

) such that the difference

d(x)=f(x)-g(x)

is an odd function; that is,

d(-x)=-d(x)

for all

x

[-a,a]

. Suppose further that the region between the graphs of

y=f(x)

and

y=g(x)

on the half‐interval

0\le x\le a

has area

9

square units. (a) Without evaluating any antiderivatives, determine the exact value of

\displaystyle\int_{-a}^{a}\bigl(f(x)-g(x)\bigr)\,dx

. Provide a complete explanation. (b) Find the total area of the region enclosed between

y=f(x)

and

y=g(x)

on the entire interval

[-a,a]

. Justify your answer by appealing to symmetry. (c) Define

h(x)=f(x)+5

and

k(x)=g(x)+5

. Determine the area of the region enclosed between

y=h(x)

and

y=k(x)

[-a,a]

, and explain why your result follows without explicit integration.