Find the area of the region lying in the first quadrant and bounded by `y=4x^2`,`x = 0, y = 1 a n d y = 4`.

Explanation

The area of the region bounded by the curve `y=4x^(2)` and the lines `y=4,x=0, y=1` in first quadrant is shown by the shaded region.

` :. ` Required area (shaded region)
`=int_(1)^(4)xdy`
`=int_(1)^(4)(1)/(2)sqrt(y) dy=(1)/(2)[(y^(3//2))/(3//2)]_(1)^(4)`
`=(1)/(3)(4^(3//2)-1^(3//2))`
`=(1)/(3)(8-1)=(7)/(3)` sq. units.