13.1 Areas: Definition, Names, and Notations
We start with the area of a rectangle with side lengths \(A\) 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 \(\int\) thing?
We use this notation because everyone else does. Cheer up! You will be able to recognize and read statements involving these symbols. They are not a threat. Imagine it as a weird S, standing for Sum. The integral is a weird kind of sum.
What is \(dx\)?
It indicates that if we divide the interval between the endpoints of the integral into tiny slivers of length \(dx\), the contribution to the area from the sliver containing the value \(x\) and it is almost a rectangle, the area of that rectangle will be the y difference between its ends, which is \(f(x)\) multiplied by the s difference which for each sliver is \(dx\); \(f(x)dx\), is thus the area in that sliver whose sum over all the slivers is the area we seek. If \(f(x)\) is \(\sin x\), The average value of \(\sin x\) here, multiplied by \(dx\) is what we sum over all the slivers to get the indicated integral.
In describing your integral you can often leave out the \(x\)’s when describing its endpoints , writing it as
Sometimes you want to describe the endpoints by giving the values of something other than the \(x\) value,
(Occasionally you may want to describe the lower and upper endpoints of the \(x\) interval by values of some other function \(g(x)\) in which you can do so by indicating the endpoints by \(g(x)=a\) and \(g(x)=b\)). For example the integral here could be described just as well as the integral from \(x^3=0\) to \(x^3=1\).)
We call this the integral from \(0\) to \(1\) of the sine function.
It is the area between the four boundaries \(x = 0, x = 1, y = 0\), and \(y = \sin(x)\) counting any area below the \(x\) axis as negative.
In general the area or integral from \(a\) to \(b\) of "the integrand" \(z(x)\) times \(dx\), is the area bounded by \(x = a, x = b, y = z(x)\) and \(y = 0\). Area below \(y = 0\) is counted negatively.
What happens if the "lower limit" \(a\) is bigger than the "upper limit" \(b\)?
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. To make this happen we define the area from bigger to smaller to be minus the area from smaller to bigger. With this definition, the integral from \(a\) to \(b\) plus that from \(b\) to \(c\) is the integral from \(a\) to \(c\) no matter the numerical order of the numbers \(a, b\) and \(c\).
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
\[g(t) = \int_0^t \sin(x)dx\]
And now we can ask, what is the derivative of the function \(g\) defined this way, as a function of \(t\)?