Question

Find the area of the region bounded by y = `sqrt(x)` and y = x.

Answer 1

We are given the equations of curve y = `sqrt(x)` and line y = x.

Solving y = `sqrt(x)`

⇒ y² = x and y = x,

We get x² = x

⇒ x² – x = 0

⇒ x(x – 1) = 0

∴ x = 0, 1

Required area of the shaded region

= `int_0^1 sqrt(x)  "d"x - int_0^1 x  "d"x`

= `2/3 [x^(3/2)]_0^1 - 1/2 [x^2]_0^1`

= `2/3[(1)^(3/2) - 0] - 1/2 [(1)^2 - 0]`

= `2/3 - 1/2`

⇒ `(4 - 3)/6`

⇒ `1/6` sq.units

Hence, the required area = `1/6` sq.units