Question

Find the number `b` such that the line y=b divides the region...?

Find the number `b` such that the line y=b divides the region bounded by the curves `y=x^2` and `y=4` into two regions with equal area.

Answer 1

Find the area of the region first.

Explanation:

The area of the region is given by
`int_-2^2(4-x^2)dx = 2int_0^2(4-x^2)dx`
`= 2(4x-x^3/3)_0^2`
`= 32/3`

Second, `y = b` intersects the curve `y = x^2` when
`x = +-sqrtb`.
Third, we want to find b such that
`int_-sqrtb^sqrtb(b-x^2)dx = 16/3`
This will occur if and only if
`int_0^sqrtb(b-x^2)dx = 8/3`
Integrate:
`int_0^sqrtb(b-x^2)dx =(bx-x^3/3)_0^sqrtb`
`= bsqrtb-(bsqrtb)/3`
`= (2bsqrtb)/3`

Now set this equal to `8/3`.
`(2bsqrtb)/3 = 8/3`
`bsqrtb = 4`
`b^(3/2) = 4`
`b = root(3)(16)`
or `b = 2root(3)(2)`