Answer to Question #327298 in Calculus for Divs

Question

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.

Answer 1

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

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

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

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

Let's find $\overline{x}$:

$y^2=4x$ , $x=1$

$y=\pm2\sqrt x$ , $0\le x\le1$

$dV=\pi y^2(x)dx=\pi (2\sqrt x)^2dx=\pi\cdot4xdx$

$\displaystyle\int_V xdV=\displaystyle\int_0^1 x\pi\cdot4xdx=\frac43\pi x^3|_0^1=\frac43\pi$

Volume can be found using the formula:

$V=\displaystyle\int_V dV=\pi\displaystyle\int_0^1y^2(x)dx=\pi\displaystyle\int_0^1(2\sqrt x)^2dx$$=\pi\displaystyle\int_0^14xdx=\pi\cdot2x^2|_0^1=2\pi$

$\overline{x}=\frac{\displaystyle\int_V xdV}{\displaystyle\int_V dV}=\displaystyle\frac43\pi/(2\pi)=\frac23$

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

$\overline{y}=0$

$\overline{z}=0$

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