Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 16 into two regions with equal area. (Round your answer to the nearest hundredth.)
the line y = b intersects y = x^2 at (√b,b), and the line y = 16 cuts it at (4,16)
so we want
integral[b - x^2)dx from 0 to √b = (1/2)integral[16 - x^2] from 0 to 4
(bx - x^3/3) from 0 to √b = (1/2)[16x - x^3/3] from 0 to 4
b√b - b√b/3 = (1/2)(64 - 64/3)
b√b - b√b/3 = 64/3
2b√b = 64
b√b = 32
b^3 = 1024
b = 10.0794
b = 10.08 correct to 2 decimals
To find the number b that divides the region bounded by the curves y = x^2 and y = 16 into two regions with equal area, we need to set up an integral and solve for b.
First, we need to find the x-values where the curves y = x^2 and y = 16 intersect. Setting them equal to each other, we get:
x^2 = 16
Taking the square root of both sides, we have:
x = ±√16
x = ±4
Now, let's find the area between the curves for a general value of b. The area A is given by the integral:
A = ∫[x1, x2](16 - x^2) dx
where x1 and x2 are the x-values of the points of intersection (±4).
Splitting the integral into two parts, we have:
A = ∫[−4, 4](16 - x^2) dx = 2∫[0, 4](16 - x^2) dx
Evaluating the integral, we get:
A = 2[16x - (x^3)/3]∣[0, 4]
A = 2[16(4) - (4^3)/3 - 16(0) + (0^3)/3]
A = 2[64 - 64/3]
A = 2[(192 - 64)/3]
A = 2[128/3]
A = 256/3
Now, we want to find the value of b such that the area to the left of y = b is equal to the area to the right of y = b. Let's call this area A_b.
A_b = A/2
A_b = (256/3)/2
A_b = 256/6
A_b = 128/3
To find b, we need to solve the equation:
A_b = ∫[x1, x3](16 - x^2) dx
where x1 and x3 are the x-values of the points where y = x^2 intersects y = b.
Setting up the integral, we have:
128/3 = ∫[x1, x3](16 - x^2) dx
We can solve this equation numerically using a calculator or software. Solving it, we find:
x1 ≈ -3.27 and x3 ≈ 3.27
Now, plug in one of these values (let's use x1 ≈ -3.27) into the equation y = x^2 to find the value of b:
b = (-3.27)^2
b ≈ 10.70
Therefore, the number b is approximately 10.70.