Question regarding line element

Question

$y=bx^2$ is a parabola. Let h be the height of a variable line segment parallel to the x-axis having it's end points on the parabola. I wish to solve: $$\int_{\sqrt{\frac{h}{b}}}^{-\sqrt{\frac{h}{b}}}(-\textbf{i})\bullet (d \textbf{l})$$

Wouldn't $d\textbf{l}=dx(-\textbf{i})$ Since we are moving in negative x direction??

Answer 1

We have:

$$\int_{\sqrt{\frac{h}{b}}}^{-\sqrt{\frac{h}{b}}}(-\textbf{i})\bullet (d \textbf{l})=-\int_{\sqrt{-\frac{h}{b}}}^{\sqrt{\frac{h}{b}}}-(\textbf{i})\bullet (d \textbf{l})= \int_{-\sqrt{\frac{h}{b}}}^{\sqrt{\frac{h}{b}}}(\textbf{i})\bullet (d \textbf{l})= \int_{-\sqrt{\frac{h}{b}}}^{\sqrt{\frac{h}{b}}}dx $$