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Q.
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
Application of Integrals
Solution:
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
$=\int\limits_{0}^{2} \left\{\left(x^{2}+1\right)-\left(4x-3\right)\right\}dx$ - Area of $\Delta AOD $
$= \int\limits_{0}^{2} \left(x^{2}-4x+4\right)dx -\frac{1}{2} \times\frac{3}{4}\times3 $
$=\left[\frac{\left(x-2\right)^{3}}{3}\right]_{0}^{2} -\frac{9}{8}=\frac{37}{24} = 1.54$
