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Q.
Determine the area under the curve $y=\sqrt{a^{2}-x^{2}}$ included between the lines $x = 0$ and $x = a$.
Application of Integrals
Solution:
We have given the equation of curve
$y=\sqrt{a^{2}-x^{2}}$
$\Rightarrow \quad y^{2}=a^{2}-x^{2}$
$\Rightarrow \quad y^{2}+x^{2}=a^{2}$
Thus the area of the shaded region
$=\int\limits_{0}^{a} \sqrt{a^{2}-x^{2}} dx=\left[\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}sin^{-1}\frac{x}{a}\right]_{0}^{a}$
$=\left[0+\frac{a^{2}}{2}sin^{-1}\left(1\right)-0-\frac{a^{2}}{2}sin^{-1}\left(0\right)\right]=\frac{\pi a^{2}}{4}$
