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?
The equations you provided are correct for finding the centroid, but the equation you used to calculate the area is incorrect. The shape described by the given equations is not a quarter of an oval/ellipse. It is actually a region between the curve of y = sqrt[x], the x-axis, and the vertical line x = 9.
To find the area of this region, you can use the integral:
Area = Integral from 0 to 9 of [y(x) - 0] dx
Since y = sqrt[x] and y = 0 in this case, this simplifies to:
Area = Integral from 0 to 9 of sqrt[x] dx
To evaluate this integral, you can use the power rule of integration, which states that the integral of x^n dx is (x^(n+1))/(n+1). Applying this rule, we get:
Area = [(2/3)x^(3/2)] evaluated from 0 to 9
= (2/3)(9^(3/2) - 0^(3/2))
= (2/3)(27 - 0)
= 18
So the correct area of this region is 18, not pi*9*3.
To find the x-coordinate of the centroid, you can use the equation:
x-coordinate = (1/Area) * Integral from 0 to 9 of (x * f(x)) dx
Substituting the values, we have:
x-coordinate = (1/18) * Integral from 0 to 9 of (x * sqrt[x]) dx
To evaluate this integral, you can use integration by parts. Let u = x and dv = sqrt[x] dx. Then, du = dx and v = (2/3)x^(3/2). Applying the integration by parts formula, we get:
x-coordinate = (1/18) * [(uv - Integral of v du)] evaluated from 0 to 9
= (1/18) * [(x * (2/3)x^(3/2)) - (Integral of (2/3)x^(3/2) dx)] evaluated from 0 to 9
= (1/18) * [(2/3)x^(5/2) - (4/15)x^(5/2)] evaluated from 0 to 9
= (1/18) * [(10/15)x^(5/2)] evaluated from 0 to 9
= (5/27) * (9^(5/2) - 0^(5/2))
= (5/27) * (9^5/2)
= (5/27) * (3^5)
= 125/3
Therefore, the x-coordinate of the centroid is 125/3 or approximately 41.67.
To find the y-coordinate of the centroid, you can use the equation:
y-coordinate = (1/Area) * Integral from 0 to 9 of [(1/2) * (f(x))^2] dx
Substituting the values, we have:
y-coordinate = (1/18) * Integral from 0 to 9 of [(1/2) * (sqrt[x])^2] dx
= (1/18) * Integral from 0 to 9 of (1/2) * x dx
= (1/18) * [(1/2)(x^2/2)] evaluated from 0 to 9
= (1/18) * [(1/4)(9^2)] - (1/18) * [(1/4)(0^2)]
= (1/18) * (81/4)
= 9/8
Therefore, the y-coordinate of the centroid is 9/8 or approximately 1.125.
So the exact coordinates of the centroid are approximately (41.67, 1.125).