We start with the area of a rectangle with sides A and B. As you know this area is AB. Our first task is to use this fact to provide a means of finding areas of irregular figures.
To do so we must first define precisely what we are trying to do.
Suppose we have some function, for example the sine function, and have an interval, say from 0 to 1, on the x axis. We can then plot the curve defined by
y(x) = sin(x)
and ask for the area in the region whose sides are: the lines x = 0, x = 2, y = 0 and the given curve.
This area is called the definite integral of the function sin(x) from lower limit 0 to upper limit 1. The word definite is sometimes left out, and the area is then called the integral from 0 to 1.
The standard notation for it is:
Why this ugly notation? Why the weird thing?
We use this notation because everyone else does. Cheer up! you will be able to recognize and read statements involving these symbols.
I guess the thing is there as an indication that you have a sort of a sum here, and things you are summing are, sort of, sin(x)dx.
What is dx?
Patience, patience. You will see.
You can also leave out the x='s that lie near the integral sign (which is the weird thing), which leaves you
We call this the integral from 0 to 1 of the sine function. And it is the area between x = 0, x = 1 y = 0 and y = sin(x) counting any area below the x axis as negative.
The more general integral is the integral from a to b of "the integrand", which is the name we give to the function that defines one boundary of our area.
And what happens if the "lower limit" is bigger than the "upper limit"?
The area between x = a and x = b plus the area between x = b and x = c is, when a is less than b and b less than c, merely the area between a and c.
This is such a wonderful property that we define the integral in the case you mention to make it hold true for all a, b and c. This means the area from a to b plus the area from b back to a must be the area from a to a, which is nothing at all. So we define the area from bigger to smaller to be minus the area from smaller to bigger.
And what good is all this?
Our key task is to figure out how to determine what these areas are. And we have a mighty tool for doing this.
First notice that the notion of integral here gives us a new way to define a function. We can make the upper limit of our integration vary, call it t, and consider the resulting integral as a function of t.
For example, we can write
And now we can ask, what is the derivative of the function g defined this way, as a function of t?