Determine the area under the curve $y = \sqrt {{a^2} - {x^2}}$ included between the lines $x$ $= 0$ and $x = a$.

We have $y = {\sqrt a ^2} - {x^2}$
$\Rightarrow$ ${y^2} = {a^2} - {x^2}$

$\Rightarrow$ ${x^2} + {y^2} = {a^2}$

Graph of above function is semi-circle lying above x-axis.
The graph is as shown in the following figure.

From the figure, area of shaded region,

$A = \int_0^a {\sqrt {{a^2} - {x^2}} } dx$
$= \left[ {\frac{x}{2}\sqrt {{a^2} - {x^2}} + \frac{{{a^2}}}{2}{{\sin }^{ - 1}}\frac{x}{a}} \right]_0^a$

$= \left[ {0 + \frac{{{a^2}}}{2}{{\sin }^{ - 1}}1 - 0 - \frac{{{a^2}}}{2}{{\sin }^{ - 1}}0} \right] = \frac{{{a^2}}}{2} \cdot \frac{\pi }{2} = \frac{{\pi {a^2}}}{4}$ sq, units