Question

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

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
\[y^2 = 2\left( y + 4 \right)\]
\[ \Rightarrow y^2 - 2y - 8 = 0\]
\[ \Rightarrow \left( y - 4 \right)\left( y + 2 \right) = 0\]
\[ \Rightarrow y = 4\text{ or }y = - 2\]
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
\[= \int_{- 2}^4 x_{\text{ line }} dy - \int_{- 2}^4 x_{\text{ parabola }} dy\]
\[ = \int_{- 2}^4 \left( y + 4 \right)dy - \int_{- 2}^4 \frac{y^2}{2}dy\]
\[ = \left.\frac{\left( y + 4 \right)^2}{2}\right|_{- 2}^4 - \left.\frac{1}{2} \times \frac{y^3}{3}\right|_{- 2}^4 \]
\[ = \frac{1}{2}\left( 64 - 4 \right) - \frac{1}{6}\left[ 64 - \left( - 8 \right) \right]\]
\[ = 30 - 12\]
\[ = 18\text{ square units }\]