标量线积分

Weighing a curve

A worked integral along a parabola

Independence of parametrization

Curves in space and higher dimensions

For the parametrization $(t, t^2)$ on $[0, 1]$, what does $ds$ become in the integral?

What is $\int_C x\, ds$ along $y = x^2$ from $(0,0)$ to $(1,1)$?

The same arc of parabola is parametrized by $(t, t^2)$ on $[0,1]$ and again by $(u^2, u^4)$ on $[0,1]$. What does the second parametrization change?

A wire along a curve has constant density $3$ grams per unit length, and the curve's arc length is $5$. Give its mass, as a plain number.

Give $\int_C (x^2 + y^2)\, ds$ over the circle of radius $2$ traversed once, in the form a*pi like 12*pi.