Question #327298

Locate the centroid of the volume bounded by the equation y^2 = 4x, x = 1 and the x-axis and revolving about the x-axis.


1
Expert's answer
2022-04-13T09:16:30-0400



The equations for calculating the centroidal coordinates of a volume of arbitrary shape are given as

x=VxdVVdV=VxdVV\overline{x}=\frac{\displaystyle\int_V xdV}{\displaystyle\int_V dV}=\frac{\displaystyle\int_V xdV}{\displaystyle V}

y=VydVVdV=VydVV\overline{y}=\frac{\displaystyle\int_V ydV}{\displaystyle\int_V dV}=\frac{\displaystyle\int_V ydV}{\displaystyle V}

z=VzdVVdV=VzdVV\overline{z}=\frac{\displaystyle\int_V zdV}{\displaystyle\int_V dV}=\frac{\displaystyle\int_V zdV}{\displaystyle V}

Let's find x\overline{x}:

y2=4xy^2=4x , x=1x=1

y=±2xy=\pm2\sqrt x , 0x10\le x\le1

dV=πy2(x)dx=π(2x)2dx=π4xdxdV=\pi y^2(x)dx=\pi (2\sqrt x)^2dx=\pi\cdot4xdx

VxdV=01xπ4xdx=43πx301=43π\displaystyle\int_V xdV=\displaystyle\int_0^1 x\pi\cdot4xdx=\frac43\pi x^3|_0^1=\frac43\pi

Volume can be found using the formula:

V=VdV=π01y2(x)dx=π01(2x)2dxV=\displaystyle\int_V dV=\pi\displaystyle\int_0^1y^2(x)dx=\pi\displaystyle\int_0^1(2\sqrt x)^2dx=π014xdx=π2x201=2π=\pi\displaystyle\int_0^14xdx=\pi\cdot2x^2|_0^1=2\pi

x=VxdVVdV=43π/(2π)=23\overline{x}=\frac{\displaystyle\int_V xdV}{\displaystyle\int_V dV}=\displaystyle\frac43\pi/(2\pi)=\frac23

No necessary to calculate y\overline{y} and z\overline{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

z=0\overline{z}=0

Answer: Coordinates of the centroid: (23,0,0)\frac23,0,0) .


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