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)#