Explanation

(a)
$\int_{0}^{10\pi}{|\sin{x|dx = 10\left\lbrack \int_{0}^{\pi/2}{\sin{xdx + \int_{\pi/2}^{\pi}{\sin{x\ dx}}}} \right\rbrack}}$
[$\because|\sin{x|}$is periodic with period $\pi$]
$= 10\lbrack\lbrack - \cos{x\left. \ \ \right\rbrack_{0}^{\pi/2} + \lbrack - \cos{x\left. \ \ \right\rbrack_{\pi/2}^{\pi}}}\rbrack$
$= 10\lbrack 1 + 1\rbrack = 20$