For me the integral problem is so interesting starting from the attempt to calculate the area of a circle, once can cut into minute rings like an onion and then stack them to form a triangle, area of which = 1/2 * r * 2pier = pie*r^2. Beautiful.

To generalize this concept, mathematicians found they can solve hard problem by summing up many many small values. Hence, the concept of derivative (df(x)/dx) is developed. In essence, it solves or reconciles approximation and precise.

is to be understood in two ways. First, it gives the change in a function F(x) between x0 and x given that its rate of change is f(x), in terms of the definite integral over f between the said limits, assuming the integral can be found as the area under the graph.

We live in a multi-dimensional world, so naturally, the math, particularly this integral calculus needs to cope with multiple variable.

An important concept is Lagrange Multiplier

Also to know that there are two systems cartisian and polar to use.