The area (in sq. units) bounded by the parabola y=x^2-1, the tangent…

We have,

$y=x^2-1⇒\frac{dy}{dx}=2x⇒(\frac{dy}{dx})_{(2,3)}=4$

The equation of the tangent to the parabola $y = x^2 -1$ at (2, 3) is $y-3=4(x-2)$ or, $4x-y-5=0$.

The required area A is given by

A = Area of the shaded region

$⇒A=\int\limits_0^2|y_2-y_1|dx$

$⇒A=\int\limits_0^2(y_2-y_1)dx=\int\limits_0^2[(x^2-1)-(4x-5)]dx$

$⇒A=\int\limits_0^2(x^2-4x+4)dx=\int\limits_0^2(x-2)^2dx=\left[\frac{(x-2)^3}{3}\right]_0^2$

$⇒A=0-(\frac{-8}{3})=\frac{8}{3}$