The equation you give in the title doesn't seem to appear in the image (which you should reproduce the relevant part as MathJax and also crop it much more aggressively). The equation you reference is ${\int }_{C}f(x,y)ds={\int }_a^{b}f(x,0)dx$ There is no $f(x)$. The domain of integration is critical here, so omitting it is omitting the key aspect. If you don't see the relevance of one side of the equation being written as ${\int }_C$ (with $ds$) and the other as ${\int }_a^b$ (with $dx$), then that's what you need to understand. Once you do, what's happening in the equation should be clear. In general, if some equation seems not to be making sense, you should pay attention to all the details on each side. Mathematical notation is dense, and small differences can hold worlds of meaning.

Derek Elkins‭ I recast my post. Better now?