Define for .

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

Problem

Define

p(x)=e^{-|x|}

for

-2\le x\le 2

. (a) Prove that

p

is an even function. (b) Express the average value of

p

[-2,2]

as a single definite integral with limits of integration from

0

2

only, and then evaluate it exactly. (c) Compute the average value of

p

[0,2]

. Compare it to the average value found in part (b) and provide a clear explanation for any difference.