Calculus

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?

If you mean the area bordered by y = sqrt x, y=0 and x=9.
The value of that area is
INTEGRAL OF: sqrt (x) dx
0 to 9
= x^(3/2)/(3/2) @ x=9 - x^(3/2)/(3/2) @ x=0
= (2/3)*27 - 0 = 18
===============================
The x-centroid Xc is
[INTEGRAL OF: x*sqrt (x) dx]/(area)
0 to 9
= [x^(5/2)/(5/2)@x=9]/ 18
Xc = (2/5)(243)/18 = 5.4
===============================
The y-centroid Yc is
[INTEGRAL OF: y*sqrt (x) dx]/(area)
0 to 9
= [INTEGRAL OF: x dx]/(area)
0 to 9
= [x^2/2]/18 @ x=9
= 2.25

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  1. The answer to the Y-centroid is incorrect.
    The answer for that part is 1.125 or (81/72)

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