Question

Find the area bounded by the curve y = `sqrt(x)`, x = 2y + 3 in the first quadrant and x-axis.

Answer 1

Given that: y = `sqrt(x)`, x = 2y + 3, first quadrant and x-axis.

Solving y = `sqrt(x)` and x = 2y + 3

We get y = `sqrt(2y + 3)`

⇒ y² = 2y + 3

⇒ y² – 2y – 3 = 0

⇒ y² – 3y + y – 3 = 0

⇒ y(y – 3) + 1(y – 3) = 0

⇒ (y + 1)(y – 3) = 0

∴ y = –1, 3

Area of shaded region

= `int_0^3 (2y + 3) "d"y - int_0^3 "y"^2  "d"y`

= `[2 y^2/2 + 3y]_0^3 - 1/3 [y^3]_0^3`

= `[(9 + 9) - (0 + 0)] - 1/3[27 - 0]`

= 18 – 9

= 9 sq.units

Hence, the required area = 9 sq.units