Question

The area below the line $y=x+1$ and bounded by the curve $x^2+y^2-4x-6y-19=0$ is

Answer 1

$C_1\Rightarrow y=x+1$

$C_1\Rightarrow x^2+y^2-4x-6y-19=0$

$(x-2{)}^2+(y-3{)}^2=32$

Method $1$

Since the given line passes through centre of the circle it divides the circle in two equal halves.

Radius of circle $\Rightarrow \sqrt{32}$ units.

Area of shaded region $\Rightarrow \frac{Ar\left(circle\right)}{2}$

$\Rightarrow \frac{\pi r^2}{2}=\frac{\pi (\sqrt{32}{)}^2}{2}=16\pi$