Question

The area of the region enclosed by the parabola x² = y, the line y = x + 2 and the x-axis, is 

Options

• ${\displaystyle \frac{2}{9}}$

• ${\displaystyle \frac{9}{2}}$

• 9

• 2

Answer 1

${\displaystyle \frac{9}{2}}$

Explanation:

y = x² is a parabola with vertex (0, 0) and axis of parabola is y-axis.

For intersection of y = x² and y = x + 2

x² = x + 2

⇒ x² – x – 2 = 0

⇒ (x – 2)(x + 1) = 0

⇒ x = – 1, 2

For y = x + 2

${\displaystyle \begin{array}{lll}x& -1& 0\\ y& 1& 2\end{array}}$

= ${\displaystyle {\int }_{-1}^2(x+2) dx-{\int }_{-1}^2 x^2 dx={\left[\frac{{(x+2)}^2}{2}\right]}_{-1}^2+{[-\frac{x^3}{3}]}_{-1}^2}$

= ${\displaystyle \frac{9}{2}}$ sq.units