How do you solve this?

Question

Find the number b such that the line y = b divides the region bounded by the curves y = 16x² and y = 16 into two regions with equal area. (Round your answer to two decimal places.)

Answer 1

Get the area of the full region first. You could integrate w respect to y after isolating x in the given eqn or, maybe easier, do 32 - ∫_-1¹ 16x^2dx. (The area you're looking for is the 2 x 16 rectangle - the area under the parabola.)

Take 1/2 that area and set up an integral = to that (let's call that 1/2 area B). This one probs is easier to do w respect to y; that way, the number you're looking for (call it k) becomes the upper bound of integration:

x = ±1/4√y so, to treat the ± , let's just use the + function and double it:

2· ∫₀^k1/4√ydy = B.

Often on the AP, they will just ask you to set this up rather than solve for k. But solving isn't hard either. You'll get an equation which reduces to k^3/2 = # then just take both sides to the 2/3 power.

Happy mathing!