Let be a continuous even function and be a continuous odd function on the symmetric…

MATHEMATICS · Subjective · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let

f

be a continuous even function and

g

be a continuous odd function on the symmetric interval

[-a,a]

where

a>0

. (i) Write the symbolic expressions for the average values

\bar f

and

\bar g

f

and

g

[-a,a]

. (ii) Without carrying out any integration, explain why

\bar g=0

. (iii) If

f(x)\ge 0

for all

x

[-a,a]

and

f

is not identically zero, compare

\bar f

with

0

, giving a brief reason.