Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area.

Answer 1

because of the symmetry, we can just go from x=0 to x=2 to find the area between
y = x^2 and y = 4

that area = ∫4-x^2 dx from 0 to 2
= [4x - (1/3)x^3] from 0 to 2
= 8 - 8/3 - 0
= 16/3

so when y = b
x= √b
and we have the area as
∫(b - x^2) dx from 0 to √b
= [b x - (1/3)x^3] from 0 to √b
= b√b - (1/3)b√b - 0

(2/3)b√b = 8/3
b√b =4
square both sides
b^3 = 16
b = 16^(1/3) = 2 cuberoot(2)
or appr 2.52