Is there any good resource for understanding the theory behind the centroid of a mass, $(\bar{x}, \bar{y})=(\cfrac1A\int_{a}^{b}xf(x)dx, \cfrac1A\int_{a}^{b}\cfrac12f(x)^2)$, particularly for someone who doesn't have much a background in physics? The proof given in my calc 2 textbook isn't pleasant to follow and I can't find any decent illustrations of a proof
I understand the concept. I get I'm supposed to imagine something like a fulcrum balancing an object at a point such that the object does not tip over. I just don't understand how to derive the formulas.
Areas make more sense as double rather than single integrals. The area of an object would simply be
$$A = \iint_A 1\cdot dA$$
and having an integrand other than one can be interpreted as being the density of some object with a mass. In this context the center of mass is given by
$$\bar{x} = \frac{1}{A} \iint_A x\cdot dA$$
$$\bar{y} = \frac{1}{A} \iint_A y\cdot dA$$
where the interpretation is that the integrand is the value that is averaged over $A$ so the integral finds the "average" $x$ and the "average" $y$, a reasonable definition for a center of mass (a centroid is purely geometric so the density is assumed to be $1$ so that the mass $=$ area).
In single variable calculus, however, you are taught that integrals get you areas under the curve of a function. This lines up because if we set up the double integral with $y$ first and try to find the area between a function and the $x$ axis
$$A = \int_{x=a}^{x=b} \int_{y=0}^{y=f(x)}1\cdot dy\:dx = \int_a^b f(x) \: dx$$
which lines up perfectly. Just take the previous double integrals further with these exact same bounds to derive the formulas from your textbook:
$$\bar{x} = \frac{1}{A} \int_a^b \int_0^{f(x)} x\:dy\:dx = \frac{1}{A} \int_a^b x\:f(x)\:dx$$
$$\bar{y} = \frac{1}{A}\int_a^b \int_0^{f(x)} y \:dy\:dx = \frac{1}{A}\int_a^b \frac{1}{2}[f(x)]^2\:dx$$