Let for real

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Let

g(\theta)=\theta\sin\theta

for real

\theta

. A researcher examines the interval

[\theta_{0},2\pi]

and wishes the average value of

g

over that interval to be exactly

1

. Parts: (a) Establish an integral expression for the average value of

g

over

[\theta_{0},2\pi]

in terms of

\theta_{0}

. Do not yet evaluate. (b) Compute

\displaystyle\int_{\theta_{0}}^{2\pi} g(\theta)\,d\theta

and simplify your result as far as possible. (c) Set your simplified average-value expression equal to

1

and derive an equation that must be satisfied by

\theta_{0}

. (d) Solve the equation from part (c) for

\theta_{0}

with

0<\theta_{0}<2\pi

. Provide exact values and discuss whether more than one solution exists in this interval. (e) Analyze the sign and monotonicity of

g(\theta)

[\theta_{0},2\pi]

to justify why the average value condition leads to the specific solution(s) you obtained.