Paramaterizing a path $C$ along a parabola $y=2x^2$

Question

I am doing a line integral where the path $C$ is defined as the arc of the parabola $y=2x^2$ from the points $(-1,2)$ to $(2,8)$.

Is there a "catch all" approach or method that can be applied here? Or is the only way to parametrize this is to think of an expression in terms of $t$ that works for a particular interval of $t$?

Answer 1

Let $x=t$. Then, if $y=f(x)$, we have the following parametrization: $$x=t;\quad y=f(t)$$ I think that this is a "catch all" approach!

Answer 2

The approach of $x=t, y=f(t)$ works for all curves that are functions of $x$.

That is, the approach fails for $y^2=x$ (although a nearly identical approach may be used).

A worse case is when it completely fails for something that is not a function of $y$ or $x$. For example, a circle of radius 1: $x^2 + y^2 = 1$. For arbitrary shapes like this, a little more thinking is required; however, the independent variable substitution works pretty well in most cases.