I know that when finding area between curves, and they're both given as functions $f\left(x\right)$ then the integral becomes something like $\int ( f_1 - f_2 ) dx$ where $f_1$ is the top function and $f_2$ is the bottom function. So I find the length of the vertical lengths
But then, if I want to check my answer by integrating with respect to $y$, then I have to change my $f\left(x\right)=y$ functions into $g\left(y\right)=x$ functions, and then find the horizontal lengths.
This might be obvious, but aren't we just finding the inverse functions?
The reason I'm asking is because I've always found integrating with respect to $y$ hard since I've either forgotten to find $g\left(y\right)=x$ or I can't find the limits of integration... but if it's just finding inverse functions, then that solving becomes so much conceptually clearer.
Yes, you are right if you feel integrating along x-axis is easier for you. It is just a game of change the name of the variable.
In the following case, the original functions are $f_1(x)=2x,f_2(x)=x^2$, there inverse functions are $g_1(x)=\frac{x}{2},g_2(x)=\sqrt x$.
$\int_0^2(f_1(x)-f_2(x))=\int_0^22x-x^2=x^2-\frac{x^3}{3}|_0^2=\frac{4}{3} \tag 1$
$\int_0^4(g_2(x)-g_1(x))=\int_0^4\sqrt x-\frac{x}{2}=\frac{2}{3}x^\frac{3}{2}-\frac{x^2}{4}|_0^4=\frac{4}{3}\tag 2$
Most people find (1) is easier than (2) in both integration itself and finding integration limits, that is one reason textbooks or teacher want us to explore both ways. You should be fine going forward with your strategy - saving us the pain to tilt our heads 90$^\circ$ :-).