Find the exact coordinates of the centroid. y = sqrt[x], y = 0, x = 9.
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Is this basically 1/4 of an oval/ellipse? If so then the area would be: pi*9*3, correct?
So the X coordinate would equal: 1/Area * Integral from 0 to 9 of (x*f(x))*dx
Which equals: (4/(27*pi))*[(2/5)(x^(5/2))] evaluated at 9 and 0 which equals: 4.584?
The Y coordinate would equal: 1/Area * Integral from 0 to 3 of (1/2)*[f(x)]^2*dx
Which equals: (4/(27*pi))*(x^2)/4 evaluated at 3 and 0 which equals: 0.955
Am I using the wrong equation for area?
I answered this already. This is about the third time you have posted the question. No, it is not an ellipse.
My apologies...I did not see your initial response. I will look more closely next time.
No problem at all! Let's go through the problem and find the exact coordinates of the centroid together.
First, let's start by finding the area of the region between the curves y = sqrt[x], y = 0, and x = 9. To do this, we can use the definite integral:
Area = ∫[a, b] (f(x) - g(x)) dx
where f(x) and g(x) are the top and bottom functions, respectively, and [a, b] is the interval over which we want to find the area.
In our case, the top function is y = sqrt[x], the bottom function is y = 0, and the interval is from x = 0 to x = 9. So the area becomes:
Area = ∫[0, 9] (sqrt[x] - 0) dx
Simplifying this integral, we get:
Area = ∫[0, 9] sqrt[x] dx
To evaluate this integral, we can use the power rule for integration. The result is:
Area = [2/3 * x^(3/2)] from 0 to 9
= (2/3 * 9^(3/2)) - (2/3 * 0^(3/2))
= 18 - 0
= 18
Therefore, the area of the region is 18 square units.
Now, let's find the x-coordinate of the centroid using the formula:
x-coordinate of centroid = (1/Area) * ∫[a, b] (x * f(x)) dx
where f(x) is the top function (y = sqrt[x]).
In our case, the x-coordinate of the centroid becomes:
x-coordinate of centroid = (1/18) * ∫[0, 9] (x * sqrt[x]) dx
To evaluate this integral, we can use the u-substitution method, letting u = sqrt[x]. The result is:
x-coordinate of centroid = (1/18) * ∫[0, 9] (u^2 * u * 2u) du
= (1/18) * 2 * ∫[0, 9] u^4 du
= (1/18) * 2 * [1/5 * u^5] from 0 to 9
= (2/18) * (1/5 * 9^5 - 1/5 * 0^5)
= (2/18) * (1/5 * 9^5)
= (2/18) * (9^5/5)
= 2/10 * 9^5
= 0.4 * 59049
= 23619.6
Therefore, the x-coordinate of the centroid is approximately 23619.6.
Next, let's find the y-coordinate of the centroid using the formula:
y-coordinate of centroid = (1/Area) * ∫[a, b] (1/2 * [f(x)]^2) dx
In our case, the y-coordinate of the centroid becomes:
y-coordinate of centroid = (1/18) * ∫[0, 9] (1/2 * [sqrt[x]]^2) dx
= (1/18) * ∫[0, 9] (1/2 * x) dx
= (1/18) * (1/2 * [1/2 * x^2]) from 0 to 9
= (1/18) * (1/4 * 9^2 - 1/4 * 0^2)
= (1/18) * (1/4 * 9^2)
= 1/72 * 81
= 1.125
Therefore, the y-coordinate of the centroid is approximately 1.125.
Hence, the exact coordinates of the centroid are (23619.6, 1.125).