Maybe this is better suited for Physics Exchange, but I have been self studying line elements and I wanted to check my understanding against the community. I tried to derive the line element for some f(x,y):
Consider the function of $y = 3\sin(x)$, I parameterized using $x=(t)$
$$ y = 3\sin(x) \to x=(t), y=3\sin(t) $$
Plugging this into arc length I get:
$$ S = \int_{0}^{1}\sqrt{(1)^2+(3\cos(x))^2} \,dt $$
From that I got the line element: $$ ds^2 = (1 + 9\cos^2(x))dx^2 $$
Is this right, or am I completely wrong here???
As discussed in the comments with @PaulSinclair the original line element I posted is close, but not quite correct. $ dx^2 $ should be $ dt^2$ , and therefore the correct line element should be:
$$ ds^2 = (1 + 9\cos^2 t)dt^2 $$