Today I started learning about line integrals and I have a doubt about the concept itself of the line integral.
As I understood, measuring the line integral was the sum of the $ds$ elements multiplied by the function. So I thought of using this concept to measure the area between the function $x$ and $x^2$ in the interval $[0, 1]$.
This Area is $\int^1_0xdx-\int^1_0x^2dx=\frac{1}{6}$
But when I compute this with the line integral method gives me something different, I thought of this Area as $$ \int_\mathcal{C}f(x)ds $$ which is the sum of the arc length element (which would be the arc length element of $x^2$) multiplied by the function. But computing this gives a different result $$ \int_\mathcal{C}f(x)ds=\int^1_0x\sqrt{1+4x^2}dx\approx0.95 $$
I am obviously misunderstanding something, but I couldn't find what it is.
I would appreciate any help
Thanks.