My textbook says that for a plate bounded by two curves, $f(x)$ and $g(x)$ for $x\in[a,b]$, given $f(x)\geq g(x)$, to find the center of mass, $(x,y)$, we need the following:
$$Density=\delta(x,y)$$ $$Mass=M=\int_a^b \delta[f(x)-g(x)]dx$$ $$x=\frac{1}{M}\int_a^b \delta x[f(x)-g(x)]dx$$ $$y=\frac{1}{M}\int_a^b \frac{\delta}{2} [f^2(x)-g^2(x)]dx$$
However now, the question given is in terms of $y$. That is, I have $f(y),g(y)$, and my density function is in terms of $y$ as well. I assume the formulas will look like, for $y\in[a,b]$,
$$Density=\delta(x,y)$$ $$Mass=M=\int_a^b \delta[f(y)-g(y)]dy$$ $$y=\frac{1}{M}\int_a^b \delta y[f(y)-g(y)]dy$$ $$x=\frac{1}{M}\int_a^b \frac{\delta}{2} [f^2(y)-g^2(y)]dy$$
What I am unsure of is whether I need to swap the formulas for the $x$ and $y$ coordinates, since I have replaced all the $x$ with $y$. Is this correct?
Also, I keep reading in the textbook the "distribution is symmetric about the $y/x$-axis. How do I determine if this is the case. Is it affected by the plane, the density function, or both?
Thank you!