# Integral Applications

• July 6th 2009, 07:22 AM
Integral Applications
Hey,

I am trying to solve this question where I need to find the y-coordinate of the center of mass. Once again, I am having trouble dealing with the integral with the maximum function.

I am assuming that to find the y coordinate, you multiply the function by y, since with only 2 integrals with two variables that is what we are supposed to do.

Any suggestions on how to solve this problem, or mainly on how to eliminate the maximum would be great!

Thanks
• July 6th 2009, 07:35 AM
malaygoel
y-co-ordinate of centre of mass=

$\frac{\int\int\int (y\rho) dxdydz}{\int\int\int(\rho) dxdydz}$

when integrating with respect to z, break the integral into two different integrals

in one integral, set the limits of y from 0 to 1
in other integral, sat the limit of y from 1 to 2.

in the first of the above max(1,y^2) will be 1
in the second of the above, it will be $y^2$
• July 6th 2009, 12:14 PM
Quote:

Originally Posted by malaygoel
y-co-ordinate of centre of mass=

$\frac{\int\int\int (y\rho) dxdydz}{\int\int\int(\rho) dxdydz}$

when integrating with respect to z, break the integral into two different integrals

in one integral, set the limits of y from 0 to 1
in other integral, sat the limit of y from 1 to 2.

in the first of the above max(1,y^2) will be 1
in the second of the above, it will be $y^2$

That makes sense. So would it basically be like a preset condition like for all y such that 0 < y < 1...the integration would look one way and then another way for greater y values?

Therefore, when you do the final integrate with respect to y, for y = 0 you would sub it into the first integrated equation, and for y = 2 you would sub into the second integrated equation..

If I misunderstood you, please let me know...and thanks again for your help!
• July 6th 2009, 07:14 PM
malaygoel
Evaluate:
$
\frac{\int_1^2\int_{y^2}^4 \int_{-y}^{\frac{-yz}{y+z}} (y\rho) dxdzdy+\int_0^1\int_{1}^4 \int_{-y}^{\frac{-yz}{y+z}} (y\rho) dxdzdy}{\int_1^2\int_{y^2}^4 \int_{-y}^{\frac{-yz}{y+z}} (\rho) dxdzdy+\int_0^1\int_{1}^4 \int_{-y}^{\frac{-yz}{y+z}} (\rho) dxdzdy}
$