The area (in sq units) of the region bounded by the curve `y = sqrtx` and the lines `y = 0, y = x - 2,` is

Options

A. `(5)/(2)`
B. `(59)/(12)`
C. `(3)/(2)`
D. `(7)/(3)`

Answer

A

Explanation

`xge0` is the region of first quadrant and fourth quadrant.
`x+yle3` is half the region of the plane divided by the line
`x+y=3` where origin lies.
`x^(2)le4y` is the region inside the parabola `x^(2)=4y`
`yge1+sqrt(x)` is the region inside the parabola `y=1+sqrt(x),` which has vertex at (0,1).
So, the required region is as shown in the following figure.

From the figure,
`"area "=overset(1)underset(0)int(1+sqrt(x))dx+overset(2)underset(1)int(3-x)dx-overset(2)underset(1)int(x^(2))/(4)dx=(5)/(2)`