center of mass: functions with respect to y

Hi. In Calc 2 I have some homework questions for calculating the moments and center of mass for planar laminae. Fortunately, the book gives the exact equations necessary for the calculations, when there are functions f(x) and g(x), within bounds a and b:

$\displaystyle M_{x} = \rho\int_a^b\frac{f(x)+g(x)}{2}[f(x)-g(x)]\,dx$

$\displaystyle M_{y} = \rho\int_a^b{x[f(x)-g(x)]\,dx}$

$\displaystyle m = \rho\int_a^b{[f(x)-g(x)]\,dx}$

$\displaystyle \overline{x} = \frac{M_{y}}{m}$

$\displaystyle \overline{y} = \frac{M_{x}}{m}$

However, how do I adjust these formulae when the functions must be functions in respect to y? For example, I have a problem where the area is bounded by

$\displaystyle x = y + 2$

$\displaystyle x = y^2$

Which involves a sideways parabola.

Re: center of mass: functions with respect to y

Okay, after trying for an hour or so to understand how the moments of mass are calculated, I think I understood it, and now this seems pretty simple. It should be...

$\displaystyle M_{y} = \rho\int_{a}^{b}\frac{f(y)+g(y)}{2}[f(y)-g(y)]\,dy$

$\displaystyle M_{x} = \rho\int_{a}^{b}y[f(y)-g(y)]\,dy$

$\displaystyle m = \rho\int_{a}^{b}[f(y)-g(y)]\,dy$

$\displaystyle \overline{x} = \frac{M_{y}}{m}$

$\displaystyle \overline{y} = \frac{M_{x}}{m}$

...where a and b are on the y axis.

Re: center of mass: functions with respect to y

That seems to be correct. You simply exchange x and y, right?

- Hollywood

Re: center of mass: functions with respect to y

It's a little more complicated than that. The definitions of $\displaystyle \overline{x}$ and $\displaystyle \overline{y}$ remain the same, except that the formulae for moment around the y axis and moment around the x axis get flipped around, and the formulae are all dependent on y instead of x. It is difficult to explain why this makes sense without drawing and illustrating a graph, but it does make sense, and it has worked well for me on several problems so far.