Problem

For $k>0$, let $D_k=\{(x,y):0\le x\le k\pi \text{ and } x\sin(x)\le y\le x\}$ and let $A_k$ be the area of $D_k$. Which of the following statement is true?

(a) $A_1=2\pi+\pi^2/2$.

(b) $A_2=4\pi+2\pi^2$.

(c) $A_3=3\pi+9\pi^2/2$.

(d) $A_4=4\pi+8\pi^2$.

(e) None of the above statement is true.