Hi Bogrune
Integral and differential calculus are two quite different fields that are quite intimately related.
Differential calculus is the study of derivatives. Derivatives can be used to find the "rate of change" of a function. For example, if you plot the functions x
2 and x
3, then you will find the latter to be a lot steeper. Derivatives can be used to make this notion rigourous. In physics, derivatives are often used to find velocity and acceleration. For example, when riding in your car, and looking at the speedometer, you're actually looking at a derivative.
Other uses of derivatives is finding slopes of functions, and actually trying to graph functions. Derivatives are also useful in finding minima/maxima of functions. As such, you can solve cool problems like: "given a rectangle with length l and width w such that the circumference is 10. How to choose l and w such that the area enclosed is maximal". This is a standard problem that can be solved by derivatives, and which would be quite hard to solve without them.
Additionaly, derivatives allow you to forget stupid formula's like -\frac{b}{2a} for the top of a parabola.
Integrals are in many ways the inverse of derivatives. Integrals allow you to calculate area and volume of things. Also, you can find lengths of curves, and average values of functions.
For example, consider the function y=x^2 on [0,1], what is the average value of this function? This is a standard problem that one can solve with integrals.
Solving integrals is quite hard however. Certainly when compared to derivatives. Taking derivatives consists of applying mechanical rules, no thinking is involved. Integrals on the other hand require a lot of creativity, and are much more fun!
Vector calculus and complex analysis use integrals and derivatives
a lot! In fact, vector calculus is the study of integrals and derivatives to more dimensions. And complex analysis is the study of derivatives and integrals in complex numbers.
Number theory uses complex analysis. Statements about prime numbers can often be translated as statements about a complex function (called the Riemann-zeta function). And as such, the statements become easier to solve. Since complex analysis uses derivatives and integrals, it becomes obvious that number theory does so as well!