Find the are of the region bounded by the curve `y=x^2 + 2` and the lines `y = x,x =0 and 3 `

Explanation

`y = x^(2) + 2 implies x^(2) = (y-2)`
Clearly , `x^(2) = (y-2)` represents an upward parabola with its vertex at A (0,2)

Also , y = x represents the straight line , making an angle of `45^(@)` with the positive direction of the x-axis .
And x = 0 is the y-axis , while x = 3 represents a ling parallel to the y-axis at a distance of 3 units from it .
Thus , the shaded region in the given figure is the required area .
`therefore` required area = (area ODCAO) - (area ODBO)
`=overset(3)underset(0)(int) (x^(2) + 2) dx - overset(3)underset(0)(int) x dx`
= `[(x^(3))/(3) + 2x ]_(0)^(3) - [(x^(2))/(2)]_(0)^(3) = (15 - (9)/(2)) = (21)/(2)` sq units .
Hence , the required area is `(21)/(2)` sq units .