Page 235 - Calculus Workbook For Dummies

Chapter 12

Who Needs Freud? Using the

Integral to Solve Your Problems

In This Chapter

Weird areas, surfaces, and volumes
L’Hôpital’s Rule
Misbehaving integrals
Other stuff you’ll never use

ow that you’re an expert at integrating, it’s time to put that awesome power to use to
Nsolve some . . . ahem . . . real-world problems. All right, I admit it — the problems you
see in this chapter won’t seem to bear much connection to reality. But, in fact, integration is
a powerful and practical mathematical tool. Engineers, scientists, and economists, among
others, do important, practical work with integration that they couldn’t do without it.

Finding a Function’s Average Value

With differentiation, you can determine the maximum and minimum heights of a function, its
steepest points, its inflection points, its concavity, and so on. But there’s a simple question
about a function that differentiation cannot answer: What’s the function’s average height?
To answer that, you need integration.

Q. What’s the average value of sinx between A. Piece o’ cake. This is a one-step problem:
0 and π?
π
# sinx dx
total area
average value= = 0
base π - 0
π
- cosx @
= π 0
1 - -
- ^ 1 1h
= π
2
=
π