The area (in sq. units) bounded by the parabola y = x 2 - 1 , the tangent at the point 2 , 3 to it and the y…

We have, $\frac{dy}{dx}=2x$

$\therefore$ Slope of tangent at $\left(2,3\right)$ is

$m=4$

$\therefore$ Equation of tangent is $y-3=4\left(x-2\right)$

$\therefore$ Point $T\equiv \left(0,-5\right)$

$\therefore$ Area of $\mathrm{\Delta }PTM=\frac{1}{2}\times 8\times 2=8$

Area of curve $PAMP={\int }_{-1}^3\sqrt{y+1}dy$

$=\frac{2}{3}{\left[{\left(y+1\right)}^{\frac{3}{2}}\right]}_{-1}^3=\frac{16}{3}$

$\therefore$ Required shaded area $=8-\frac{16}{3}=\frac{8}{3}$