Question

Find the area bounded by the parabola, y = x^2-1, the tangent at the point (2,3) and y axis.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given curve is

$\rm \: y = {x}^{2} - 1$

Let we first find the equation of tangent to the above curve at A(2, 3).

So,

$\rm \: \dfrac{dy}{dx} = 2x$

$\rm \: \bigg(\dfrac{dy}{dx}\bigg)_{(2,3)} = 2 \times 2 = 4$

So, it means, slope of tangent at A(2, 3) is 4.

So, Equation of tangent line which passes through the point (2, 3) and having slope 4 is given by

$\rm \: y - 3 = 4(x - 2)$

$\rm \: y - 3 = 4x - 8$

$\rm \: y = 4x - 8 + 3$

$\rm \: y = 4x - 5$

Now, We have to find the area between the curves

$\rm \: y = 4x - 5 - - - - (1)$

$\rm \: y = {x}^{2} - 1- - - - (2)$

$\rm \: x = 0 - - - - (3)$

[ See the attachment for area to be evaluated ]

Required area is

$\rm \: =  \:\displaystyle\int_{ - 5}^{3}\rm x_{line} \: dy \: - \: \displaystyle\int_{ - 1}^{3}\rm x_{parabola} \: dy$

$\rm \: =  \:\displaystyle\int_{ - 5}^{3}\rm \frac{y + 5}{4} \: dy \: - \: \displaystyle\int_{ - 1}^{3}\rm \sqrt{y + 1} \: dy$

$\rm \: =  \: \dfrac{1}{4} \bigg[\dfrac{ {y}^{2} }{2} + 5y\bigg]_{ - 5}^{3} - \dfrac{2}{3}({\bigg(y + 1\bigg) }^{\dfrac{3}{2} }\bigg)_{ - 1}^{3}$

$\rm \: =  \: \dfrac{1}{4} \bigg[\dfrac{ 9 }{2} + 15 - \dfrac{25}{2} + 25 \bigg]- \dfrac{2}{3}{\bigg(3 + 1\bigg) }^{\dfrac{3}{2} }$

$\rm \: =  \: \dfrac{1}{4} \bigg[40 - 8 \bigg] - \dfrac{2}{3}{\bigg(4\bigg) }^{\dfrac{3}{2} }$

$\rm \: =  \: \dfrac{1}{4} \bigg[32 \bigg] - \dfrac{2}{3}{\bigg( {2}^{2} \bigg) }^{\dfrac{3}{2} }$

$\rm \: =  \:8 - \dfrac{16}{3}$

$\rm \: =  \:\dfrac{24 - 16}{3}$

$\rm \: =  \:\dfrac{8}{3} \: square \: units$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$