The Riemann Integral
Riemann integrals of functions are defined and the mean value theorem for integrals is proved. The relationship between integration and differentiation is uncovered in the fundamental theorem of calculus. The techniques of integration by substitution and by parts are established, and the latter is used to develop Stirling’s formula for the approximation of the factorial of a positive integer. It is also used to prove the irrationality of the constants π and e. Integrals are employed to define the concept of arc length of a curve and this idea is then used to give a geometric interpretation of π. Methods for approximating definite integrals are discussed. The chapter concludes with an introduction to improper integrals and the integral test for convergence of a series.