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 $P = \{x_0, x_1, \ldots, x_n\}$ and sample points $\xi_i \in [x_{i-1}, x_i]$:

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

If, as the mesh $|P| = \max_i (x_i - x_{i-1})$ tends to $0$, the sums $S(f,P)$ converge to the same value $I$ regardless of how the sample points $\xi_i$ are chosen, then $f$ is Riemann integrable on $[a,b]$ and we set $\int_a^b f(x)\,dx = I$.

Fubini's Theorem

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

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

Mere Riemann integrability of $f$ on $D$ does not guarantee that the inner integral exists for every $x$. (Example: the function equal to $1$ exactly when $x = 1/2$ and $y$ is rational is integrable on $D$, yet the inner integral at $x = 1/2$ does not exist.) The more general statement is treated from the intermediate level onwards.

Change of Variables Formula

For a $C^1$ map $T$ that is one-to-one on the interior and whose Jacobian does not vanish, writing $(x,y) = T(u,v)$:

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

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, the convergence and divergence of improper integrals (infinite intervals and discontinuous functions), the basics of multivariable integration (multiple integrals) with Fubini's theorem, and change of variables with the Jacobian. Computational techniques such as substitution and integration by parts are covered in the introductory course and are assumed here; the basic level treats the theoretical foundations of integration and its extension to several variables.

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?

A bounded function is Riemann integrable exactly when its set of discontinuities has measure zero (Lebesgue's criterion). The Riemann integral is therefore limited to functions that are close to continuous, and the Dirichlet function, which is discontinuous at every point, is not Riemann integrable. The Lebesgue integral, based on measure theory, has the flexibility 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 the monotone convergence theorem it can handle difficult problems in analysis.