Asked by Anonymous
Find the equation of the horizontal line that divides the area of the region in half
y=18−x^2, y=0.
y=18−x^2, y=0.
Answers
Answered by
Steve
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
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
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