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 :

• A. $\frac{37}{24}$
• B. $\frac{8}{3}$
• C. $\frac{187}{24}$
• D. $\frac{14}{3}$

Answer 1

The correct option is A $\frac{37}{24}$

$y=x^2+1$
$\frac{dy}{dx}=2x$
$\frac{dy}{dx}|(2,5)=4$
$\therefore$ Equation of tangent at $(2,5)$ is
$y-5=4(x-2)$
$\Rightarrow 4x-y=3$
$\Rightarrow \frac{x}{3/4}-\frac{y}{3}=1$



Area of the shaded region
$=\underset{0}{\overset{2}{\int }}(x^2+1)dx-\text{ar}\left(△APB\right)$
$={[\frac{x^3}{3}+x]}_0^2-\frac{1}{2}\times \frac{5}{4}\times 5$

$=\frac{8}{3}+2-\frac{25}{8}$

$=\frac{37}{24}$