Rectifiable curves in $\mathbb{R}^n$ have measure $0$, but line integral is nonzero.

Question

I covered a bit of Lebesgue measure and came across the property that states that the integral of a function $f$ over some nullset $E$ is always $0$. If rectifiable curves have measure $0$ in $\mathbb{R}^n$ , how come every line integral is not vanishing?

I know this is very elementary and I must be making a big mistake somewhere. I am trying to get the grasp of Lebesgue measure/some intuition for this problem. Any help is appreciated!

Answer 1

You are confusing one and two dimensional measures. The line segment joining $(0,0)$ to $(1,0)$ in the plane has area (planar Lebesgue measure) $0$ but still has length $1$. Line integrals along curves use the arclength along the curve as the underlying measure.