Answer
The area is $A = \frac{2}{3}$.
Work Step by Step
We have
$x\left( t \right) = \sin t$, $x'\left( t \right) = \cos t$,
$y\left( t \right) = {\cos ^2}t$.
Using Eq. (11) the area is
$A = \mathop \smallint \limits_{t = 0}^{\pi /2} {\cos ^2}t\left( {\cos t} \right){\rm{d}}t = \mathop \smallint \limits_{t = 0}^{\pi /2} {\cos ^3}t{\rm{d}}t$.
Using Eq. (6) of Section 8.1 we get
$A = \mathop \smallint \limits_{t = 0}^{\pi /2} {\cos ^3}t{\rm{d}}t = \frac{1}{3}{\cos ^2}t\sin t|_0^{\pi /2} + \frac{2}{3}\mathop \smallint \limits_{t = 0}^{\pi /2} \cos t{\rm{d}}t = \frac{1}{3}{\cos ^2}t\sin t|_0^{\pi /2} + \frac{2}{3}\sin t|_0^{\pi /2}$
$A = \frac{2}{3}$.
