The area of the region bounded by the parabola ( y - 2 ) ^ 2 = ( x...

Given, the parabola ${\left(y - 2\right)}$^{2} = \left(x - 1\right) and y = 3, $x = 2$

So, point is $( 2, 3 )$ Differentiate given equation w.r.t. $x$, we get$</p><p>$2 ( y - 2 ) {y}^{'} = 1$</p><p>${y}^{′} = $\dfrac{1}{2 ( y - 2 )}</p><p>${{y}^{′}|}_{( 2, 3 )} =\dfrac{1}{2}

So,$the equation of tangent at\left(2, 3\right)</p><p>$y - 3 = $\dfrac{1}{2}\left(x - 2\right)</p><p>$x - 2y + 4 = 0$</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/0a31cad0-b18a-48b6-a436-3fd3607fd023-image.png" style="width: 337px; height: 276px;" />$

Required area $= {∫}_{0}^{3} [ ( y - 2{)}^{2} + 1 ] dy - {∫}_{0}^{3} ( 2y - 4 ) dy$
Required area $= {∫}_{0}^{3}\left({y}^{2} - 4y + 5\right)$dy + {∫}_{0}^{3}\left(2y - 4\right)$dy$
Required area $= {\left[\dfrac{{y}^{3}}{3} - 2{y}^{2} + 5y\right]}_{0}^{3} - {\left[{y}^{2} - 4y\right]}$_{0}^{3}$</p><p>Required area$= 9$sq units$