Question

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

• A. $\frac{\pi a^2}{4}$
• B. $\frac{\pi a^3}{4}$
• C. $\frac{\pi a^2}{8}$
• D. none of these

Answer 1

Answer: (A)

We have given the equation of curve
$y=\sqrt{a^2-x^2}$
$\Rightarrow y^2=a^2-x^2$
$\Rightarrow y^2+x^2=a^2$

Thus the area of the shaded region
$=\underset{0}{\overset{a}{\int }}\sqrt{a^2-x^2}dx={\left[\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}si{n}^{-1}\frac{x}{a}\right]}_0^a$
$=\left[0+\frac{a^2}{2}si{n}^{-1}\left(1\right)-0-\frac{a^2}{2}si{n}^{-1}\left(0\right)\right]=\frac{\pi a^2}{4}$