IN THIS CHAPTER we discuss the Riemann on a finite interval \([a,b]\), and improper integrals in which either the function or the interval of integration is unbounded.
- SECTION 3.1 begins with the definition of the Riemann integral and presents the geometrical interpretation of the Riemann integral as the area under a curve. We show that an unbounded function cannot be Riemann integrable. Then we define upper and lower sums and upper and lower integrals of a bounded function. The section concludes with the definition of the Riemann–Stieltjes integral.
- SECTION 3.2 presents necessary and sufficient conditions for the existence of the Riemann integral in terms of upper and lower sums and upper and lower integrals. We show that continuous functions and bounded monotonic functions are Riemann integrable.
- SECTION 3.3 begins with proofs that the sum and product of Riemann integrable functions are integrable, and that \(|f|\) is Riemann integrable if \(f\) is Riemann integrable. Other topics covered include the first mean value theorem for integrals, antiderivatives, the fundamental theorem of calculus, change of variables, integration by parts, and the second mean value theorem for integrals.
- SECTION 3.4 presents a comprehensive discussion of improper integrals. Concepts defined and considered include absolute and conditional convergence of an improper integral, Dirichlet’s test, and change of variable in an improper integral.
- SECTION 3.5 defines the notion of a set with Lebesgue measure zero, and presents a necessary and sufficient condition for a bounded function \(f\) to be Riemann integrable on an interval \([a,b]\); namely, that the discontinuities of \(f\) form a set with Lebesgue masure zero.