Question

Find the area of the region bounded by the parabola y² = 2x and the straight line x − y = 4. [NCERT EXEMPLAR]

Answer 1

The parabola y² = 2x opens towards the positive x-axis and its focus is $\left(\frac{1}{2},0\right)$.

The straight line x − y = 4 passes through (4, 0) and (0, −4).

Solving y² = 2x and x − y = 4, we get

$$\begin{gathered} y^2=2\left(y+4\right) \\ \Rightarrow y^2-2y-8=0 \\ \Rightarrow \left(y-4\right)\left(y+2\right)=0 \\ \Rightarrow y=4\mathrm{or}y=-2 \end{gathered}$$

So, the points of intersection of the given parabola and the line are A(8, 4) and B(2, −2).



∴ Required area = Area of the shaded region OABO

$$\begin{gathered} ={\int }_{-2}^4{x}_{\mathrm{line}}dy-{\int }_{-2}^4{x}_{\mathrm{parabola}}dy \\ ={\int }_{-2}^4\left(y+4\right)dy-{\int }_{-2}^4\frac{y^2}{2}dy \\ ={\overline{)\frac{{\left(y+4\right)}^2}{2}}}_{-2}^4-\frac{1}{2}\times {\overline{)\frac{y^3}{3}}}_{-2}^4 \\ =\frac{1}{2}\left(64-4\right)-\frac{1}{6}\left[64-\left(-8\right)\right] \\ =30-12 \\ =18\mathrm{square}\mathrm{units} \end{gathered}$$