Question

Find the area of the region bounded by \[y = \sqrt{x}\] and y = x.

Answer 1

The curve\[y = \sqrt{x}\] or \[y^2 = x\]  represents a parabola opening towards the positive x-axis. 

The curve y = x represents a line passing through the origin.
Solving \[y^2 = x\] and y = x, we get \[x^2 = x\]
\[ \Rightarrow x^2 - x = 0\]
\[ \Rightarrow x\left( x - 1 \right) = 0\]
\[ \Rightarrow x = 0\text{ or }x = 1\]
Thus, the given curves intersect at O(0, 0) and A(1, 1).

∴ Required area = Area of the shaded region OAO 
\[= \int_0^1 y_{\text{ parabola }} dx - \int_0^1 y_{\text{ line }} dx\]
\[ = \int_0^1 \sqrt{x}dx - \int_0^1 xdx\]
\[ = \left.\frac{x^\frac{3}{2}}{\frac{3}{2}}\right|_0^1 - \left.\frac{x^2}{2}\right|_0^1 \]
\[ = \frac{2}{3}\left( 1 - 0 \right) - \frac{1}{2}\left( 1 - 0 \right)\]
\[ = \frac{2}{3} - \frac{1}{2}\]
\[ = \frac{1}{6}\text{ square units }\]