So I'm struggling if I should interpret the integral as a sum of infinitesimally small quantities or as just the antiderivative. I know these two things are equivalent but I can't think of those two things at the same time when I'm doing an integral. The reason I find this troublesome is because it's giving a headache to mentally transform all the integrals I see into antiderivatives, instead of just thinking of it as an operation that adds differentials. Specifically I'm calculating work done by variable forces along non-straight lines using integrals. If I consider these integrals as just sums then they're no big deal. But if I see them as antiderivatives of some mysterious function then it gives me a headache. What do you think about this? What's your interpretation of the integral? If I don't think of it as an antiderivative then the limits of my integral seem (although not entirely) somewhat arbitrary. Thanks EDIT: Also, in a simple fashion: why does Leibniz notation allow algebra-like manipulations, and provide valid statements that way?