Question

The area (in sq. units) in the first quadrant bounded by the parabola , $y=x^2+1$, the tangent to it at the point $(2,5)$ and the coordinate axes is

Answer 1

Answer: 1.54

The equation of parabola $x^2=y-1$
The equation of tangent at $(2,5)$ to parabola is
$y-5={\left(\frac{dy}{dx}\right)}_{\left(2,5\right)}\left(x-2\right)$
$y-5=4(x-2)$
$4x-y=3$
Then, the required area
$=\underset{0}{\overset{2}{\int }}\left\{\left(x^2+1\right)-\left(4x-3\right)\right\}dx$ - Area of $\Delta AOD$
$=\underset{0}{\overset{2}{\int }}\left(x^2-4x+4\right)dx-\frac{1}{2}\times \frac{3}{4}\times 3$
$={\left[\frac{{\left(x-2\right)}^3}{3}\right]}_0^2-\frac{9}{8}=\frac{37}{24}=1.54$