Question

The area of the region (in square units) bounded by the curve x² = 4y, line x = 2 and x-axis is
(a) 1
(b) 2/3
(c) 4/3
(d) 8/3

Answer 1

(b) 2/3



Point of intersection of the parabola x² = 4y and straight line x = 2 is given by
$$\begin{gathered} x^2=4y\mathrm{and}x=2 \\ \Rightarrow 4=4y \\ \Rightarrow y=1 \\ A\left(2,1\right)\mathrm{is}\mathrm{the}\mathrm{point}\mathrm{of}\mathrm{intersection}\mathrm{of}\mathrm{the}\mathrm{curve}\mathrm{and}\mathrm{straight}\mathrm{line} \\ \mathrm{Area}\mathrm{of}\mathrm{shaded}\mathrm{region}OAB={\int }_0^{2}ydx \\ ={\int }_0^2\frac{x^2}{4}dx \\ ={\left[\frac{x^3}{12}\right]}_0^2 \\ =\frac{2^3}{12}-0 \\ =\frac{2}{3}\mathrm{square}\mathrm{units} \end{gathered}$$