TABLE OF CONTENTS Now we turn to the idea of integration, which gives us a way to add up infinitely many things, each of which is infinitesimally small. Actually doing the addition is not only not recommended, it is not even possible unless we use some approximations along the way. And that is the purpose of integration, to find the limits of infinite sums exactly and without the tedium of actually doing infinite sums. We saw a version of this earlier using the Greek method of exhaustion. (Computing the area of a shape that is not a rectangle is a good example of the adding up of nearly infinitely many nearly infinitesimally thin rectangles, or triangles in the case of polygons approximating a circle.)
We will need a good, efficient way to express sums of a large number of terms, usually an unspecified number n, such as
To avoid the trouble of writing out such expressions in full, we can use a single symbol. The symbol 
means that we are to substitute i = 1, 2, 3, , n successively in the expression following it and then add the results. Here are three examples,
The numbers or letters below and above ? tell us where to start and stop summing. They are called the limits of the summation (not to be confused with the "limit" of a sequence or function). There is nothing special about the letters i, j, and
