Question

What will be 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.
(a) $\dfrac{14}{3}$
(b) $\dfrac{187}{24}$
(c) $\dfrac{37}{24}$
(d) $\dfrac{8}{3}$

Answer 1

Hint: To solve this we will draw the graph first with the help of the given equations then we will find the area bounded in the first quadrant by integrating the curve in the first quadrant and then subtracting any excess area, if any exists.

Complete step-by-step answer:
To solve this question we will need to draw the figure using the given equations to see the given problem in a more clear way. But before that we will find the equation of the tangent to the parabola $y={{x}^{2}}+1$ at the point (2, 5).
To find the equation of the tangent at a point to the parabola we have a shortcut method where, we can replace,
$\begin{align}
  & {{x}^{2}}\to x{{x}_{1}} \\
 & y\to \dfrac{y+{{y}_{1}}}{2} \\
\end{align}$
Where \[\left( {{x}_{1}},{{y}_{1}} \right)\] is the point where we have to find the tangent.
So, we get the tangent at (2, 5), as
\[\begin{align}
  & \left( \dfrac{y+5}{2} \right)=x.2+1 \\
 & \Rightarrow y+5=4x+2 \\
 & \Rightarrow y=4x-3 \\
\end{align}\]

So we get the equation of the tangent at (2, 5) as: $y=4x-3$.
When we put y = 0, we get x = $\dfrac{3}{4}$.

So,
area of the shaded region = (area under the parabola from x = 0 to x = 2) – (area of the triangle formed by the three shown coordinates)
we get,
area of the shaded region = \[\int_{0}^{2}{\left( {{x}^{2}}+1 \right)dx-\dfrac{1}{2}\times \left( 2-\dfrac{3}{4} \right)\times 5}\]
$\begin{align}
  & =\left[ \dfrac{{{x}^{3}}}{3}+x \right]_{0}^{2}-\dfrac{25}{8} \\
 & =\left( \dfrac{8}{3}+2 \right)-\dfrac{25}{8} \\
\end{align}$
Taking the LCM and further solving, we get
$=\dfrac{37}{24}$
Hence our answer matches with option (c)

So, the correct answer is “Option c”.

Note: We can also solve for the equation of the tangent by assuming the equation of the tangent as $y=mx+c$ and then find the values of m and c using the point given i.e. (2, 5) and using the equation of the parabola. Most of the students forget to subtract the excess area from the obtained area and end up with the wrong answer so you need to be careful about it.