"University calculus"-chape5-integral Method--the definition of integral

This chapter discusses the definition of integrals and the basic theorem of calculus.

The author previously in the Mathematical Proof column on the Gauss theorem in the beginning of the proof, given a section on the calculus of thought summary, the text mentioned in accordance with the definition of derivative (differential), according to its inverse definition to give the definition of integral and calculation method, here is actually and not rigorous, the integral itself has its own definition, And the computational method is what the basic theorem of calculus presents.

Definition of Integral:

The essence of the modern definition of integral is the Riemann sum, the author's previous introduction to the definition of multiple integrals has already been mentioned, here is the definition of one-dimensional integrals, the relative two-weight, Sanchong integral will be much simpler.

The theory always stems from the actual problem, we introduce the integral in solving the problem of the curved side trapezoid area which the curve and the coordinate system surround.

You can see that the area of the curved trapezoid can be approximated in the following form:

n tends to infinity and is no longer approximate equal but strictly equal. The integral symbol is introduced here.

The strict definition of calculus is as follows:

A detailed explanation of the integral symbol:

By definition we are able to obtain the following operational properties:

Fully understanding the integral concept is very important for the right application of calculus, one of its specific manifestations is that in a series of physical problems, the discretization of the equation has not been able to adapt to our exploration of complex problems, here we need to take an integral variable DX lists the existence of integral equation to solve, And a full understanding of the concept will also be conducive to a series of high-level applications of calculus-differential geometry, partial differential, differential equations, etc. play a central role.

"University calculus"-chape5-integral Method--the definition of integral