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

$\frac{\pi a^2}{4}$ square units

The correct answer is Option (1) → $\frac{\pi a^2}{4}$ square units

Given equation of the curve is $y = \sqrt{a^2 - x^2}$.

$\Rightarrow y^2 = a^2 - x^2 \Rightarrow y^2 + x^2 = a^2$

$∴$ This is the equation of circle with centre $(0, 0)$ and radius '$a$'.

$∴$ Required area of shaded region $= \int\limits_{0}^{a} \sqrt{a^2 - x^2} \, dx$

$= \left[ \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} \right]_{0}^{a}$

$\left[ ∵\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} \right]$

$= \left[ 0 + \frac{a^2}{2} \sin^{-1}(1) - 0 - \frac{a^2}{2} \sin^{-1} 0 \right]$

$= \frac{a^2}{2} \cdot \frac{\pi}{2} = \frac{\pi a^2}{4} \text{ sq. units}$