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
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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
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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
The answer to the Y-centroid is incorrect.
The answer for that part is 1.125 or (81/72)
The exact coordinates of the centroid are (5.4, 2.25).
To find the exact coordinates of the centroid, we first need to calculate the area bounded by the curves y = sqrt(x), y = 0, and x = 9. The area can be found by taking the integral of sqrt(x) with respect to x from 0 to 9.
The integral of sqrt(x) dx from 0 to 9 is given by:
∫(0 to 9) sqrt(x) dx = [2/3 * (x^(3/2))] from 0 to 9
= (2/3 * (9^(3/2))) - (2/3 * (0^(3/2)))
= (2/3 * 27) - 0
= 18
So, the area bounded by the curves is 18.
Next, we can find the x-coordinate of the centroid. The x-coordinate of the centroid, denoted as Xc, can be calculated by taking the integral of x * sqrt(x) with respect to x from 0 to 9 and dividing it by the area.
The integral of x * sqrt(x) dx from 0 to 9 is given by:
∫(0 to 9) x * sqrt(x) dx = [2/5 * (x^(5/2))] from 0 to 9
= (2/5 * (9^(5/2))) - (2/5 * (0^(5/2)))
= (2/5 * 243) - 0
= 486/5
= 97.2
To find the x-coordinate of the centroid, we divide the result by the area:
Xc = (97.2) / 18
= 5.4
So, the x-coordinate of the centroid is 5.4.
Finally, let's find the y-coordinate of the centroid, denoted as Yc. The y-coordinate of the centroid can be calculated by taking the integral of y * sqrt(x) with respect to x from 0 to 9 and dividing it by the area.
The integral of y * sqrt(x) dx from 0 to 9 is given by:
∫(0 to 9) sqrt(x) * sqrt(x) dx = ∫(0 to 9) x dx
= [1/2 * (x^2)] from 0 to 9
= 1/2 * (9^2) - 1/2 * (0^2)
= 1/2 * 81
= 40.5
To find the y-coordinate of the centroid, we divide the result by the area:
Yc = (40.5) / 18
= 2.25
So, the y-coordinate of the centroid is 2.25.
Therefore, the exact coordinates of the centroid are (5.4, 2.25).