Let $h(x)=\begin{cases}\sin x, & 0\le x\le\dfrac{\pi}{2},\\[6pt]1, & \dfrac{\pi}{2}<x\le\pi,\\[6pt]\cos x, & \pi<x\le\dfrac{3\pi}{2}.\end{cases}$ …

Mathematics

Problem

Let

h(x)=\begin{cases}\sin x, & 0\le x\le\dfrac{\pi}{2},\\[6pt]1, & \dfrac{\pi}{2}<x\le\pi,\\[6pt]\cos x, & \pi<x\le\dfrac{3\pi}{2}.\end{cases}

Answer the following questions. a. (i) List the three sub-intervals that cover

[0,\tfrac{3\pi}{2}]

and state the corresponding expression for

h(x)

on each sub-interval. b. (ii) Form a single expression involving definite integrals that represents the average value of

h(x)

[0,\tfrac{3\pi}{2}]

. c. (iii) Evaluate the integrals obtained in (ii) and hence determine the exact average value of

h(x)

[0,\tfrac{3\pi}{2}]

. Simplify your answer.