By now we are mostly done with the theory part of definite integration. Before we dive into all the techniques, here is a sum up of the Riemann sum and the idea of cutting things into pieces then add up.
You have seen in Math 151 (or any Cal 1 class you took) how to define a tangent line at point P(x0,y0): you start off computing a secant line at point P with another point Q(x,y) on the curve. You let the point Q vary, and as Q gets closer and closer to P, we define the tangent line as the limiting result. This kind of “approximate then take the limit” idea is the same essential idea when we define definite integrals.
The first questions we see in Math 152 is computing the area under curve. We will again start with approximation, then take the limit. We cut the region under curve into n vertical strips (n being a large integer), approximate the area of each strip by a rectangle (of course, you can choose to use trapezoid, or whatever else whose area is easy to compute), and add up all the areas to get the approximate area of the whole region. By now, the approximation step is over. Taking the limit in this case is making n really large–the larger n is, the thinner the strips are, the more accurate the approximation is. The limit is then defined to be the area under curve.
This kind of “cut things up, approximate piece by piece and take the limit” works for the definition of all cases, as long as the object you consider is a product. An example is defining the displacement. We know if the velocity is constant, then the displacement between time t1 and t2 is simply velocity*(t2-t1). What if the velocity is not constant? We zoom in in time until the time interval is so small, it looks like the velocity is almost constant (like Neo in the Matrix. Remember how he dodged all the bullets? Apply that skill here), we approximate the displacement in that small time interval by the product of a sample velocity (say, the velocity at the end of the small time interval) and the length of the interval. Add all of these small displacements up to obtain an approximation to the total displacement between t1 and t2, then we take the limit: limit is taken so that each approximation is good, which means the small time intervals are really small. Finally, we define this result to be the displacement.
This would be a consistent theme throughout the course. Later we will learn how to compute volumes and work, one from analytical geometry and one from physics. How did people come up with these fancy formulas and computations? It is all the same idea we have talked about in Section 5.1: cut things up into small pieces, approximate it then take the limit.