Chapter Review

This chapter, "Integral Calculus and Its Uses," has much the same place in our study of integration as Chapter 4, "Differential Calculus and Its Uses," had in our study of differentiation. In both chapters we study significant applications and gather together the important techniques of calculation.

The major applications in this chapter are at the beginning and near the end. In the beginning we studied the problem of determining the center of mass of a complicated object. The subdivide-and-conquer strategy enabled us to extend summation formulas for discrete calculations to integral formulas for the corresponding continuous problem. Near the end of the chapter we used definite integral calculations to represent complicated periodic functions as sums of basic sine and cosine functions.

The computer algebra system Sage is a quick and powerful computational device for evaluating integrals — both numerically and symbolically. In order to understand procedures for numerical evaluation of integrals, we investigated various numerical integration procedures, from the crude right- and left-hand sums to the more efficient Midpoint and Trapezoidal Rules, and then to the weighted average of these last two, Simpson's Rule. Sage uses a more sophisticated method than Simpson's Rule to determine numerical values of definite integrals.

Although Sage will generate antiderivatives for most common functions, it is sometimes necessary or desirable to carry out integral calculations by hand. Important techniques for doing this are algebraic substitutions and trigonometric substitutions for the variable of integration.

Another powerful tool for calculating integrals is integration by parts. Both substitution and integration by parts are inverses of differentiation rules: Substitution is the inverse of the Chain Rule, and integration by parts is the inverse of the Product Rule.

Finally, we explored the symbolic computational power of Sage for finding antiderivatives. There are often situations in which, in principle, some combination of algebraic manipulation, substitution, integration by parts, and other techniques would permit pencil-and-paper calculation of an indefinite integral, but it would be extremely tedious or difficult to do so. This is where a CAS can be an extremely powerful tool. But effective use of that tool sometimes requires human interaction as well, for example, in arranging an input in a form that allows the CAS to function properly. The one human characteristic that can't be programmed into a CAS is judgment.