I've come across many instances where sometimes the error tends to zero but other times it does not. Let me give you a few examples. 1. When I calculate the volunme of a sphere summing up discs of height dy from -R to +R, the error in volume tends to zero as Delta y->0 but when I'm calculating the surface area using rings of height dy, this error does not tend to zero. Same with a hollow cone vs a solid cone. 2. ds is the arc length. We know ds/dt = speed = |dr/dt| since the error tends to zero. However, in another case, particularly from Irodov's problems: 3. I'm trying to find the work done by the spring as the block moves from one end to the other(Yes' I'm aware there are easier ways of going about it) Here, where I think I'm going wrong is assuming the spring force to be constant in the interval dy when it can only remain constant during an infinitesimal displacement along the spring. However, here we assume pressure to be constant in the interval Rdtheta when it really only is constant in the interval dH since it's a function of h And here, we assume the potential energy of the chain to be constant in the interval Rdtheta when it should be only constant in dh yet in these two cases the error tends to zero but it doesn't in the first case.