Find the equation of the horizontal line that divides the area of the region in half
y=18−x^2, y=0.
we can use symmetry and only work with the right half of the region. We could use vertical strips of width dx, but then we'd have to change boundaries where the line intersects the parabola.
So, let's use horizontal strips of height dy and length x, where x=√(18-y). Thus, we want to find k such that
∫[0,k] √(18-y) dy = ∫[k,18] √(18-y) dy
36√2 - 2/3 (18-k)^(3/2) = 2/3 (18-k)^(3/2)
4/3 (18-k)^(3/2) = 36√2
(18-k)^(3/2) = 27√2
18-k = 9∛2
k = 18-9∛2 ≈ 6.66
a good approximation would thus be
y = 20/3
To find the equation of the horizontal line that divides the area of the region in half, we first need to find the area of the region.
The given region is bound by the curve y = 18 - x^2 and the x-axis in the interval where 0 ≤ x ≤ 3. To find the area between the curve and the x-axis, we can integrate the function y = 18 - x^2 with respect to x over this interval:
Area = ∫[0,3] (18 - x^2) dx
Let's calculate this integral to find the area.
∫(18 - x^2) dx = [18x - (1/3)x^3] evaluated from 0 to 3
= (18(3) - (1/3)(3)^3) - (18(0) - (1/3)(0)^3)
= (54 - 9) - (0 - 0)
= 45
Therefore, the area of the region is 45 square units.
Since we want to find a horizontal line that divides this area in half, we need to locate the x-coordinate where the area on one side of the line is half of the total area.
The area to the left of a given x-coordinate is obtained by integrating the function over the interval [0, x]. So, we want to find the x-coordinate at which the following equation holds true:
∫[0,x] (18 - x^2) dx = 1/2 * Area
Substituting the calculated area value, we have:
∫[0,x] (18 - x^2) dx = 1/2 * 45
Simplifying, we obtain:
∫[0,x] (18 - x^2) dx = 22.5
To find x, we need to solve this equation. We can rearrange the equation as follows:
18x - (1/3)x^3 = 22.5
Multiply both sides by 3 to eliminate the fraction:
54x - x^3 = 67.5
Rearranging again, we have a cubic equation:
x^3 - 54x + 67.5 = 0
Solving this equation can be done using numerical methods or a graphing calculator. By solving this equation, we can find the x-coordinate where the area is divided in half. Once we have this x-coordinate, we can substitute it into the equation of the curve y = 18 - x^2 to find the corresponding y-coordinate and thus determine the equation of the horizontal line that divides the area in half.
Hence, the equation of the horizontal line that divides the area of the region in half is dependent on the x-coordinate found by solving the cubic equation x^3 - 54x + 67.5 = 0, and the corresponding y-coordinate is obtained by substituting this x into the equation y = 18 - x^2.