# Estimate the area between the curve y2 = x and the lines x = 0 and x = 2.

Solution:

Given Curves are

y2 = x         — (i)

x = 0          — (ii)

x = 2          — (iii)

Clearly (i) is a parabola that has a verticex at (0, 0) and the line of symmetry is y = 0 with x $$\geq$$ 0.

Also, (ii) and (iii) are the straight lines that are parallel to y-axis. With these information, the sketch in figure, in which the bounded region by (i), (ii), (iii) is the shaded portion.

Clearly the region has two symmetrical parts I and II. From figure, part – I, x moves from 0 to 2.

Now, the area of the bounded region is

$$A = 2\left | \int_{x=0}^{2} y dx \right |$$

= $$2\left | \int_{0}^{2} \sqrt{x} dx \right |$$

= $$2\left [ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right ]^2_0$$

= $$\frac{4}{3} (2\sqrt{2})$$

= $$\frac{8\sqrt{2}}{3}$$

Thus, the area of the region is $$\frac{8\sqrt{2}}{3}$$ sq. units