The equations for calculating the centroidal coordinates of a volume of arbitrary shape are given as
x ‾ = ∫ V x d V ∫ V d V = ∫ V x d V V \overline{x}=\frac{\displaystyle\int_V xdV}{\displaystyle\int_V dV}=\frac{\displaystyle\int_V xdV}{\displaystyle V} x = ∫ V d V ∫ V x d V = V ∫ V x d V
y ‾ = ∫ V y d V ∫ V d V = ∫ V y d V V \overline{y}=\frac{\displaystyle\int_V ydV}{\displaystyle\int_V dV}=\frac{\displaystyle\int_V ydV}{\displaystyle V} y = ∫ V d V ∫ V y d V = V ∫ V y d V
z ‾ = ∫ V z d V ∫ V d V = ∫ V z d V V \overline{z}=\frac{\displaystyle\int_V zdV}{\displaystyle\int_V dV}=\frac{\displaystyle\int_V zdV}{\displaystyle V} z = ∫ V d V ∫ V z d V = V ∫ V z d V
Let's find x ‾ \overline{x} x :
y 2 = 4 x y^2=4x y 2 = 4 x , x = 1 x=1 x = 1
y = ± 2 x y=\pm2\sqrt x y = ± 2 x , 0 ≤ x ≤ 1 0\le x\le1 0 ≤ x ≤ 1
d V = π y 2 ( x ) d x = π ( 2 x ) 2 d x = π ⋅ 4 x d x dV=\pi y^2(x)dx=\pi (2\sqrt x)^2dx=\pi\cdot4xdx d V = π y 2 ( x ) d x = π ( 2 x ) 2 d x = π ⋅ 4 x d x
∫ V x d V = ∫ 0 1 x π ⋅ 4 x d x = 4 3 π x 3 ∣ 0 1 = 4 3 π \displaystyle\int_V xdV=\displaystyle\int_0^1 x\pi\cdot4xdx=\frac43\pi x^3|_0^1=\frac43\pi ∫ V x d V = ∫ 0 1 x π ⋅ 4 x d x = 3 4 π x 3 ∣ 0 1 = 3 4 π
Volume can be found using the formula:
V = ∫ V d V = π ∫ 0 1 y 2 ( x ) d x = π ∫ 0 1 ( 2 x ) 2 d x V=\displaystyle\int_V dV=\pi\displaystyle\int_0^1y^2(x)dx=\pi\displaystyle\int_0^1(2\sqrt x)^2dx V = ∫ V d V = π ∫ 0 1 y 2 ( x ) d x = π ∫ 0 1 ( 2 x ) 2 d x = π ∫ 0 1 4 x d x = π ⋅ 2 x 2 ∣ 0 1 = 2 π =\pi\displaystyle\int_0^14xdx=\pi\cdot2x^2|_0^1=2\pi = π ∫ 0 1 4 x d x = π ⋅ 2 x 2 ∣ 0 1 = 2 π
x ‾ = ∫ V x d V ∫ V d V = 4 3 π / ( 2 π ) = 2 3 \overline{x}=\frac{\displaystyle\int_V xdV}{\displaystyle\int_V dV}=\displaystyle\frac43\pi/(2\pi)=\frac23 x = ∫ V d V ∫ V x d V = 3 4 π / ( 2 π ) = 3 2
No necessary to calculate y ‾ \overline{y} y and z ‾ \overline{z} z . Since the body is obtained by rotation around the x axis, it means that it is symmetrical along the x axis and the centroid point will lie on the x axis. Therefore we can conclude
y ‾ = 0 \overline{y}=0 y = 0
z ‾ = 0 \overline{z}=0 z = 0
Answer: Coordinates of the centroid: (2 3 , 0 , 0 ) \frac23,0,0) 3 2 , 0 , 0 ) .
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