How do I calculate center of mass using integrals and density?

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To calculate the center of mass using integrals and density, the x-bar is determined by the equation \bar{x} = ∫\tilde{x}dm / ∫dm, where \tilde{x} is the product of the x-coordinate and the function value, in this case, x^3. The function x^3 should be included in the integrand for the mass element dm. The density ρ should be factored in front of each integrand, affecting both the x and y calculations. The same method applies for calculating \tilde{y}, maintaining consistency in the approach. Understanding these steps is crucial for accurate center of mass calculations.
aaronfue
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Just some things I need to verify:

When finding the x-bar of the function y=x3 using the equation:

\bar{x} = \frac{∫\tilde{x}dm}{∫dm},

Is my \tilde{x} going to be the x distance(which will just be "x") times x3?

Also, will x3 be in the denominator just before "dm"?

The same process is done for the \tilde{y}, am I correct?

If I am also given a density \rho, will that stay in front of each integrand or in front of (\frac{∫\tilde{x}dm}{∫dm})?

Is the same process is done for the \tilde{y} or are there differences?

Thanks!
 
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