About half of calculus deals with area (integration), but most precalculus textbooks barely mention the subject.
Chapter 3 presents a clean and well-motivated approach to e and the natural logarithm. This approach uses the area (intuitively defined) under portions of the curve y = 1/x.
A similar approach to e and the natural logarithm is common in calculus courses. However, this approach is not usually adopted in precalculus textbooks. Using obvious properties of area, the simple presentation in this book shows how these ideas can come through clearly without the technicalities of calculus or the messy notation of Riemann sums. Students who have seen the approach given here should be well prepared to deal with these concepts in their calculus courses.
The approach taken in this book also has the advantage that it easily leads, as shown in Chapter 3, to the approximation ln(1 + h) ≈ h for small values of h. Furthermore, the same methods show that if r is any number, then (1 + r/x)x ≈ er for large values of x. A final bonus of this approach is that the connection between continuously compounding interest and e becomes a nice corollary of natural considerations concerning area.
Appendix A reviews key concepts concerning area.