Question

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

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

Answer 1

Answer: (C)

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

$=2{\left[\frac{x}{2}\sqrt{9-x^2}+\frac{9}{2}si{n}^{-1}\frac{x}{3}\right]}_1^3$

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

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

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