Question

Find the area of the region enclosed by the curves y² = x, x = `1/4`, y = 0 and x = 1, using integration.

Answer 1

The area of the region bounded by the curve,  y² = x, the line x = `1/4` and x = 4 and y = 0 (i.e., x-axis) is the a ABEF

Thus, area of ABEF = 2 area of ABCD

= `2int_(1/4)^1 ydx`

= `2int_(1/4)^1 sqrt(x)dx`

= `2[x^(3/2)/(3/2)]_(1/4)^1`

= `4/3[(1)^(3/2) - (1/4)^(3/2)]`

= `4/3[1 - 1/8]`

= `4/3[7/8]`

= `7/6` units