Consider the function

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

Problem

f(x)=e^{-x}\sin x

. Parts: (a) Derive a closed-form expression for

\displaystyle\int_{0}^{b} f(x)\,dx

in terms of

b

. Show all integration steps. (b) Using your result from part (a), obtain an explicit formula for the average value of

f

on the interval

[0,b]

. (c) Determine all positive values of

b

satisfying

0<b\le4\pi

for which the average value of

f

over

[0,b]

equals

0

. List each value of

b

in exact form and justify that no other such values occur in the stated range. (d) Explain why each value of

b

found in part (c) gives an average value of zero, relating your reasoning to the behavior of

f

over successive half-periods.