The Riemann Integral

Part 2, Chapter 2: Single-Variable Real Analysis

Learning objectives

Define the Riemann integral via lower and upper sums and the supremum-infimum criterion
State conditions guaranteeing Riemann integrability (continuous, monotone, finitely-many discontinuities)
Apply linearity, monotonicity, and the interval-additivity properties of the integral
Contrast Riemann with Lebesgue via the Dirichlet function counterexample

The integral is calculus' answer to "accumulated change". Where the derivative tells you the rate at which something is happening NOW, the integral tells you the total amount that has happened by accumulating those rates over an interval. Riemann's 1854 formalisation, partitioning the interval, forming upper and lower rectangle sums, and squeezing them together, is the standard definition for one-dimensional calculus, and although Lebesgue's broader integral has since become the working tool of analysis, every probability density, every signal energy, every solution to a differential equation begins with the Riemann concept of "area under a curve".

Partitions, upper and lower sums

A partition $P$ of $[a,b]$ is a finite list $a=x_0

The Riemann integral

If $\sup_P L(f,P)=\inf_P U(f,P)$ , the common value is called the Riemann integral $\int_a^b f(x)\,dx$ abf(x),dx, and $f$ is said to be Riemann integrable. Equivalently, $f$ is integrable iff for every $\epsilon>0$ there is a partition $P$ with $U(f,P)-L(f,P)<\epsilon$ . Continuous functions are integrable (uniform continuity does the work); monotone functions are integrable; functions with finitely many discontinuities are integrable. The Dirichlet function (1 on rationals, 0 on irrationals) is NOT Riemann integrable on $[0,1]$ , every interval contains both, so $U=1$ and $L=0$ for every partition.

Properties

The Riemann integral is linear: $\int(\alpha f+\beta g)=\alpha\int f+\beta\int g$ . It is monotone: $f\leq g\Rightarrow\int f\leq\int g$ . And it has the interval-additivity property: $\int_a^b f=\int_a^c f+\int_c^b f$ abf=intacf+intcbf for any $c\in[a,b]$ . These properties are what make the integral algebraically pleasant, you can split, combine, and bound integrals using rules that look exactly like the corresponding rules for finite sums.

The series-summer above visualises Riemann sums in their original form: partial sums of $f(x_i^*)\Delta x_i$ . Try setting $f(x)=x^2$ on $[0,1]$ with $n=10, 100, 1000$ subintervals. The lower sum (left endpoints), upper sum (right endpoints), and midpoint sum should all converge to the same limit $1/3$ . The visible gap between upper and lower sum $U-L$ shrinks like $1/n$ for monotone $f$ , that is the integrability test in action.

Where this shows up

Probability & expected values: the expectation $E[X]=\int x\,f_X(x)\,dx$ of a continuous random variable is literally an integral. Bayesian inference, statistics, and risk analysis all run on integrals of probability densities.
Signal energy & physics: the energy in a signal $x(t)$ is $\int |x(t)|^2\,dt$ . Work done by a force is $\int F\cdot ds$ . Charge passed through a circuit is $\int I(t)\,dt$ . Every accumulated quantity in engineering is an integral.
Numerical quadrature: Trapezoidal rule, Simpson's rule, and Gauss-Legendre quadrature are all explicit Riemann-type sums chosen to converge faster than the naive midpoint rule. NASA's reentry trajectories, weather simulations, and CFD all rely on numerical integration of differential equations.

Pause and think: Why does the Dirichlet function (1 on rationals, 0 on irrationals) fail to be Riemann integrable, even though it is bounded? Which step of the upper-sum, lower-sum comparison breaks down?

Try it

Predict first: compute the lower and upper Riemann sums for $f(x)=x$ on $[0,1]$ using $n=4$ equal subintervals. The true integral is $1/2$ , does your $L$ and $U$ bracket it?
Compute $\int_0^2 (3x+1)\,dx$ 02(3x+1),dx from the definition by taking a uniform partition with $n$ subintervals, computing the right-endpoint Riemann sum, and taking $n\to\infty$ . Compare with the FTC answer $\int_0^2(3x+1)dx = 8$ 02(3x+1)dx=8.
True or false: if $f$ is bounded on $[a,b]$ and continuous except at finitely many points, then $f$ is Riemann integrable. Justify.
Show that if $m\leq f(x)\leq M$ on $[a,b]$ and $f$ is integrable, then $m(b-a)\leq\int_a^b f\leq M(b-a)$ abfleqM(b−a).

A trap to watch for

The Riemann integral is not as general as people sometimes assume. Functions that are "merely bounded and defined" need NOT be integrable, the Dirichlet function is the canonical counter-example. Lebesgue's theory was invented precisely to handle this gap: it integrates a wider class of functions and behaves better under limits. For an undergraduate analysis course, Riemann is the right tool; for measure theory, probability, and modern PDE, Lebesgue is essential.

What you now know

You can build the Riemann integral from upper and lower sums, list which functions are integrable, apply the linearity and monotonicity rules, and identify pathological cases like Dirichlet. Section §2.5 (FTC) then closes the loop: it tells you how to compute integrals without falling back to Riemann sums, the antiderivative provides a shortcut.

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References

Garrity, T. (2002). All the Mathematics You Missed. Cambridge University Press, ch. 2.
Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill, ch. 6.
Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer, ch. 7.
Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley, ch. 7.
Royden, H. L., Fitzpatrick, P. M. (2010). Real Analysis (4th ed.). Pearson, ch. 2 (for the Lebesgue contrast).