Page 175 - Calculus Workbook For Dummies

Chapter 9

Getting into Integration

In This Chapter
Reconnoitering rectangles
Trying trapezoids
Summing sigma sums
Defining definite integration

n this chapter, you begin the second major topic in calculus: integration. With integration
Iyou can find the total area or volume of weird shapes that, unlike triangles, spheres,
cones, and other basic shapes, don’t have simple area or volume formulas. You can use inte-
gration to total up other things as well. The basic idea is that — with the magic of limits —
the thing you want the total of is cut up into infinitesimal pieces and then the infinite
number of pieces are added up. But before moving on to integration, you warm up with
some easy stuff: pre-pre-pre-calc — the area of rectangles.

By the way, despite the “kid stuff” quip, much of the material in this chapter and the first sec-
tion of Chapter 10 is both more difficult and less useful than what follows it. If ever there was
a time for the perennial complaint — “What is the point of learning this stuff?” — this is it.
Now, some calculus teachers would give you all sorts of fancy arguments and pedagogical
justifications for why this material is taught, but, let’s be honest, the sole purpose of teaching
these topics is to inflict maximum pain on calculus students. Well, you’re stuck with it, so
deal with it. The good news is that this material will make everything that comes later seem
easy by comparison.

Adding Up the Area of Rectangles: Kid Stuff

The material in this section — using rectangles to approximate the area of strange shapes —
is part of every calculus course because integration rests on this foundation. But, in a sense,
this material doesn’t involve calculus at all. You could do everything in this section without
calculus, and if calculus had never been invented, you could still approximate area with the
methods described here.