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.

1 Answer
Sep 19, 2017

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)