Find the area of the region bounded by and y = x.

y = √x

squaring both sides

⇒ y² = x

In y² = x parabola it is not defined for negative values of x hence the parabola will be to the right of Y-axis passing through (0, 0)

Now y = √x means y and x both has to be positive hence both lie in 1^st quadrant hence y = √x will be part of y² = x which is lying only in 1^st quadrant

And y = x is a straight line passing through origin

We have to find area between y = √x and y = x shown below

To find intersection point of parabola and line solve parabola equation and line equation simultaneously

Put y = x in y² = x

⇒ x² = x

⇒ x = 1

Put x = 1 in y = x we get y = 1

Hence point of intersection is (1, 1)

⇒ area between parabolic curve and line = area under parabolic curve – area under line …(i)

Let us find area under parabolic curve

⇒ y = √x

Integrate from 0 to 1

Now let us find area under straight line y = x

y = x

Integrate from 0 to 1

Using (i)

⇒ area between parabolic curve and line = = unit²

Hence area bounded is unit²