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Integral Calculus Explained in 5 Minutes

Integral Calculus Explained in 5 Minutes

Integral calculus is one of the two branches of calculus, the other is differential calculus. Most mathematicians refer to both branches together as simply calculus.

Calculus is all about functions, so there's no point in studying calculus until you understand the idea of a function.

Also, you should study differential calculus before integral calculus because the two have an elegant relationship with each other and this is best explained if you understand differential calculus first.

There are several ways to introduce integral calculus. The method below is a bit abstract, but it's very fast, and it also highlights the elegant relationship between integral and differential calculus. Here it is..

Suppose you have two functions g(x) and f(x) such that when you differentiate g(x) you get f(x), like this..

Dg(x)=f(x) where D is the differentiation operation "take the derivative of".

Then g(x) is called the integral of f(x) and is written as g(x)=Sf(x) where S is the integration operation "take the integral of".

Let's try an example..

Suppose I give you the function f(x)=x^2 and ask you to find a function g(x) such that when you differentiate g(x) you get f(x). The answer is..

g(x)=(x^3/3)+c where c is any constant

Try it. If you differentiate g(x) you get f(x), so the integral of x^2 is (x^3/3)+c

The elegant relationship is that differentiation and integration are "complementary" or "opposite" operations. In other words..

SDf(x)=f(x)+c where c is any constant, and..


SD means doing the two operations in succession.. take the derivative (the D operation) and then integrate the result (the S operation).

If we subtract the two results we get..


So the operation [SD-DS] reduces any function to a constant! This is the "complimentary" relationship between integration and differentiation. The special operator [SD-DS], is called the "commutator" of S and D.

By the way, the concept of a commutator is not specific to calculus, it's a general concept that applies to many mathematical operations.

The derivative of f(x) at point x=a is just the slope of the curve at that point. Does integration have a simple interpretation? Yes it does..

If we have two points x=a and x=b and we calculate the integral of f(x) at each point, then the difference of these two values is just the area under the f(x) curve between a and b. So differentiation is slope and integration is area.

Much of integral calculus is simply finding ways to integrate functions. This can get a bit boring, so why bother? Because the integral of a function is a really useful thing for solving all sorts of problems.

Content written and posted by Ken Abbott