Integration Basic

Basic (Undergraduate years 1-2)

Riemann, Improper, and Multiple Integrals

Overview

At the basic level, we study the foundations of the theory of integration in university mathematics. Whereas the introductory course focuses on computational techniques, from here on we treat integration rigorously: the precise definition of the Riemann integral, improper integrals involving infinite intervals or points of discontinuity, multiple integrals of functions of several variables, and change of variables (the Jacobian).

Learning Goals

Understand the definition of the Riemann integral and integrability conditions
Determine the convergence or divergence of improper integrals
Compute double and triple integrals
Apply change of variables and compute Jacobians

Chapter 1 Riemann Integral
Partitions, Riemann sums, integrability
Chapter 2 Improper Integral
Infinite intervals, singularities, convergence tests
Chapter 3 Double Integral
Definition, iterated integrals, order of integration
Chapter 4 Change of Variables
Jacobian, polar coordinates, cylindrical coordinates
Chapter 5 Triple Integral
Volume, mass, spherical coordinates
Chapter 6 Applications of Integration
Centroid, moment of inertia, gamma function

Average Value of a Function — explains the formula for the mean value of a function, $\frac{1}{b-a}\int f$, at a basic level
Trigonometric Substitution — explains the technique of integration by trigonometric substitution
Weierstrass Substitution — explains the Weierstrass substitution $t=\tan(x/2)$ for integrating trigonometric functions
Shell Method — explains how to find the volume of a solid of revolution by the shell method
Washer Method — explains how to find the volume of a solid of revolution by the washer method
Pappus's Centroid Theorem — explains Pappus's centroid theorem
Area in Polar Coordinates — explains the polar-coordinate area formula $A = \frac{1}{2}\int r^2\,d\theta$

Prerequisites

Content of Integration Introduction
Differentiation of functions of several variables (partial derivatives)
Basics of determinants

Basic Concepts

Riemann Sum

For a partition $P = \{x_0, x_1, \ldots, x_n\}$:

$$S(f, P) = \displaystyle\sum_{i=1}^{n} f(\xi_i)(x_i - x_{i-1})$$

The limit as the partition is refined is the Riemann integral.

Fubini's Theorem

If $f$ is integrable on $D = [a,b] \times [c,d]$, then:

$$\iint_D f(x,y)\,dA = \displaystyle\int_a^b \left(\displaystyle\int_c^d f(x,y)\,dy\right)dx = \displaystyle\int_c^d \left(\displaystyle\int_a^b f(x,y)\,dx\right)dy$$

Change of Variables Formula

When $(x,y) = T(u,v)$:

$$\iint_D f(x,y)\,dx\,dy = \iint_{T^{-1}(D)} f(T(u,v)) \left|\dfrac{\partial(x,y)}{\partial(u,v)}\right| du\,dv$$

The Jacobian $\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|$ is the local factor by which area is magnified.

Reading Corner

Differentiation and Integration Were Twins — The Bridge of the Fundamental Theorem [Reading] — Why are integration, which finds area, and differentiation, which finds slope, inverse operations? Told through history and intuition, the miracle brought about by the fundamental theorem of calculus.
Measuring Pi by Tossing a Needle — Buffon's Needle and Integration [Reading] — The wonder that $\pi$ appears merely by dropping a needle on the floor, told from the viewpoint that "probability = average over all cases = integral," tracing back to the origins of the Monte Carlo method.
The Trumpet You Cannot Paint — Finite Volume, Infinite Surface Area [Reading] — The horn-shaped solid formed by revolving $y=1/x$ has finite volume yet infinite surface area. This famous paradox is told from the viewpoint of convergence and divergence of improper integrals.

Frequently Asked Questions

What do you learn in basic integral calculus?

You learn the definition of the Riemann integral and integrability conditions, basic integration techniques (substitution, integration by parts, integration of rational functions), convergence and divergence of improper integrals (infinite intervals and discontinuous functions), and the basics of multivariable integration (multiple integrals) together with Fubini's theorem. Both computational techniques and theoretical foundations are covered.

How does basic-level integration differ from the introductory course?

The introduction focuses on computational techniques for indefinite and definite integrals (substitution, integration by parts, and so on). The basic level takes these for granted and treats the rigorous definition of the Riemann integral and integrability conditions, the convergence and divergence of improper integrals, and multiple integrals of several variables together with Fubini's theorem and change of variables (the Jacobian). What distinguishes it from the introduction is that it goes beyond computation into the theoretical foundations of integration.

What is the limitation of the Riemann integral compared with the Lebesgue integral?

The Riemann integral is limited to functions that are close to continuous (bounded, with a set of discontinuities of measure zero). Functions such as the Dirichlet function, whose discontinuities are dense, are not Riemann integrable. The Lebesgue integral, based on measure theory, adopts the flexible principle that changing the values of a function on a set of measure zero does not change the integral, and through the dominated convergence theorem and monotone convergence theorem it can handle difficult problems in analysis.