Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area.

because of the symmetry, we can just go from x=0 to x=2 to find the area between

y = x^2 and y = 4

that area = ∫4-x^2 dx from 0 to 2
= [4x - (1/3)x^3] from 0 to 2
= 8 - 8/3 - 0
= 16/3

so when y = b
x= √b
and we have the area as
∫(b - x^2) dx from 0 to √b
= [b x - (1/3)x^3] from 0 to √b
= b√b - (1/3)b√b - 0

(2/3)b√b = 8/3
b√b =4
square both sides
b^3 = 16
b = 16^(1/3) = 2 cuberoot(2)
or appr 2.52

To 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, we first need to determine the limits of integration.

Let's begin by graphing the two curves y = x^2 and y = 4:

The curve y = x^2 is a parabola that opens upwards and passes through the point (0,0). The curve y = 4 is a horizontal line that intersects the y-axis at y = 4.

Now, to find the limits of integration, we need to find the x-values where the two curves intersect.

Setting the two equations equal to each other, we have:

x^2 = 4

Taking the square root of both sides, we have:

x = ±2

So the two curves intersect at x = -2 and x = 2.

To find b, we need to find the y-value where the line y = b intersects the curve y = 4.

Since the line y = b is horizontal, it intersects the curve y = 4 at y = b.

Now, we need to find the area of the region bounded by the curves y = x^2 and y = 4.

This can be done by integrating the difference between the upper curve (y = 4) and the lower curve (y = x^2) over the limits of integration.

The equation for the area, A, is given by:

A = ∫[a,b] (f(x) - g(x)) dx

where a and b are the limits of integration, and f(x) and g(x) are the upper and lower curves respectively.

In this case, f(x) = 4 and g(x) = x^2, so the equation becomes:

A = ∫[-2,2] (4 - x^2) dx

Integrating this equation will give us the area bounded by the curves.

Using either numerical computation methods or evaluating the integral analytically, we can find the value of the integral:

A = [4x - (x^3 / 3)] evaluated from -2 to 2

After evaluating the integral, we find that A = 32/3.

Since we want the line y = b to divide the region into two regions of equal area, we need to set the area A equal to half of its value:

A/2 = 32/3 * 1/2 = 16/3

Now, we need to solve the equation for b:

16/3 = ∫[-2,2] (4 - x^2) dx

We can now solve this equation for b by integrating and setting it equal to 16/3:

16/3 = [4x - (x^3 / 3)] evaluated from -2 to b

Evaluating the integral again, we have:

16/3 = 4b - (b^3 / 3) - (-8 + 8/3)

Simplifying the equation, we get:

16/3 = 4b - (b^3 / 3) + 24/3

Multiplying through by 3, we have:

16 = 12b - b^3 + 24

Rearranging the equation, we have:

b^3 - 12b + 8 = 0

This is a cubic equation that can be solved using numerical or algebraic methods.

Using a numerical solver, we find that one of the possible solutions is b ≈ 2.7374.

Therefore, 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 is approximately 2.7374.

To find the value of b such that the line y = b divides the region into two equal areas, we first need to find the points of intersection between the curves y = x^2 and y = 4.

Setting the two equations equal to each other, we have:

x^2 = 4

Taking the square root of both sides, we get:

x = ±2

Therefore, the points of intersection are (-2, 4) and (2, 4).

To find the area between the curves from x = -2 to x = 2, we integrate the upper curve minus the lower curve:

Area = ∫[from -2 to 2] (4 - x^2) dx

Using the definite integral, we can solve for the area:

Area = [4x - (x^3/3)] [from -2 to 2]
= [4(2) - (2^3/3)] - [4(-2) - (-2^3/3)]
= [8 - 8/3] - [-8 + 8/3]
= 24/3 - 16/3
= 8/3

Since we want to divide this area into two equal parts, we need to find the line y = b such that the area between the curves from x = -2 to x = 2 is equal to 8/3.

Half of the area is equal to (8/3) / 2 = 4/3.

To find the value of b, we set up the following equation:

∫[from -2 to b] (4 - x^2) dx = 4/3

Integrating both sides gives us:

[4x - (x^3/3)] [from -2 to b] = 4/3

Simplifying the equation, we have:

[4b - (b^3/3)] - [4(-2) - (-2^3/3)] = 4/3

Expanding and simplifying further:

4b - (b^3/3) + 16 + 8/3 = 4/3

Multiplying through by 3 to clear the fraction:

12b - b^3 + 48 + 8 = 4

Rearranging the terms:

-b^3 + 12b + 52 = 4

-b^3 + 12b + 48 = 0

At this point, we can use numerical methods such as graphing or iterative techniques to find the value of b that satisfies this equation. We can also utilize the Rational Root theorem to test potential rational solutions. However, solving this equation analytically is quite challenging due to the cubic term.

Therefore, using numerical methods would be more suitable to find the value of b.