## THE INTEGRALS

All right. By now, you presumably have checked out everything about Riemann sums. It's time to move onto the formal part-- integration. First, familiarize yourself with the below symbol. It's an elongated 'S'. Why? S stands for sum. Recall the last 'Sum' you have studied.

## THE DEFINITE INTEGRAL

Now, we shall move on to the Definite integral. This is just basically what we prepped you for in Riemann sums. The following is the definition:

So, the definite integral of f(x) from a to b is just its area under the curve.

If the curve is below the x axis, the area is negative.

Note that up to now, there appears to be no relationship whatsoever between the definite and indefinite integral. How can this be? After all, they are both integrals. Soon, you shall see.

If the curve is below the x axis, the area is negative.

Note that up to now, there appears to be no relationship whatsoever between the definite and indefinite integral. How can this be? After all, they are both integrals. Soon, you shall see.

## THE INDEFINITE INTEGRAL

First, we learn something called the Indefinite integral. It is noted by the sign above.

The definition is very simple:

It's the opposite operation of the derivative. And what does this mean? That means: Differentiate F(x), you get its derivative g(x). Now, the indefinite integral of g(x) is just F(x)+C. C is a constant.

As you probably has seen, easy as it seems, it involves many hidden catches-- You are trying to reverse a process. Now, it's the power of deduction, not induction at work. And deduction? I hate it. You will Hate it too. Soon Enough. Can't believe how Sherlock Holmes lived long enough to be made into a book...

To start with, the definite integral is not a function. It's a set of functions. It should have the from of something of x plus some constant. Why? The constant turns zero when it's differentiated, thereby eliminating the difference between f(x)+c and f(x)+2c when they are integrated.

The definition is very simple:

It's the opposite operation of the derivative. And what does this mean? That means: Differentiate F(x), you get its derivative g(x). Now, the indefinite integral of g(x) is just F(x)+C. C is a constant.

As you probably has seen, easy as it seems, it involves many hidden catches-- You are trying to reverse a process. Now, it's the power of deduction, not induction at work. And deduction? I hate it. You will Hate it too. Soon Enough. Can't believe how Sherlock Holmes lived long enough to be made into a book...

To start with, the definite integral is not a function. It's a set of functions. It should have the from of something of x plus some constant. Why? The constant turns zero when it's differentiated, thereby eliminating the difference between f(x)+c and f(x)+2c when they are integrated.