What is the difference between integrating with respect to dS and dx in the context of scalar field line integrals?

Question

I have a question regarding the difference when integrating with respect to arc length and with respect to the x-axis. For example, if f(x,y) = xy and the curve, C = $x^2 + y^2$, then $\int_1^0 f(x,y)dx$ along C gives a value of $\frac{-1}{3}$. However, integrating across the same curve C, $\int_{t=0}^{t = \frac{\pi}{2}} f(x,y)dSx$ = $\frac{1}{2}$. My question is what does the value of the integral wrt to x mean? I understand that for dS, the value of the integral is equal to the area of the "curtain" (as Sal from Khanacademy puts it) under the function, defined by the curve C. However, in terms of integrating with respect to dx, I cannot understand or see what the value of the integral represents? I hope my question is clear, and cheers in advance for the help!

Answer 1

If you integrate with respect to $x$, the value of the integral represents the area of the projection of the curtain on the $y=0$ plane.