Question

Area of the smaller region bounded by $x^{2} + y^{2} = 9$ and the line $x = 1$ is

• A. $\left(2-3 \,sec^{-1}3\right)$ sq. units
• B. $\left(\sqrt{8}-3\, sec^{-1}3\right)$ sq. units
• C. $\left(9 \,sec^{-1}3-\sqrt{8}\right)$ sq. units
• D. $\left(sec^{-1}3-3\sqrt{8}\right)$ sq. units

Answer 1

Answer: (C)

We have, $x^{2}+y^{2}=9 \quad\ldots\left(i\right)$
a circle with centre $(0,0)$ and radius $3$ and line $x = 1\quad\left(ii\right)$
Required area = area of shaded region
$A=2\left[\int\limits_{1}^{3}\sqrt{9-x^{2}} dx\right]$

$=2\left[\frac{x}{2}\sqrt{9-x^{2}}+\frac{9}{2}sin^{-1}\frac{x}{3}\right]_{1}^{3}$

$=2\left[\frac{9}{2}sin^{-1}\frac{3}{3}-\frac{1}{2}\sqrt{8}-\frac{9}{2}sin^{-1}\frac{1}{3}\right]$

$=-\sqrt{8}+9\left(sin^{-1}-sin^{-1}\frac{1}{3}\right)$

$=\left(9 sec^{-1}3-\sqrt{8}\right)$ sq. units