Question

What is the area of the region bounded between the curves y=x and y=sqrt(x)?

Answer 1

`(1)/(6)`

Explanation:

y = x .....(1)

`y=√(x)` .... (2)

By solving equation (1) and equation (2)

`x = √(x)`

`√(x)\left ( √(x)-1 \right )=0`

`√(x)=0` or
`\left ( √(x)-1 \right )=0`

x = 0, x = 1

y = 0, y = 1

A =
`\int_(0)^(1)ydx(curve)-\int_(0)^(1)ydx(line)`

A =
`\int_(0)^(1)√(x)dx-\int_(0)^(1)xdx`

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

A =
`(2)/(3)-(1)/(2)`

A =
`(1)/(6)`