The fundamental theorem of calculus is the bridge that joins the two halves of the subject, and its two faces are the most quoted results in elementary mathematics. The first face says that if a function is continuous then the function obtained by accumulating it from a fixed starting point has derivative equal to the original integrand, so that differentiating an area undoes integration. The second face says that if a function has a continuous derivative then its total change over an interval is the integral of that derivative, which is the net version of the fundamental theorem of physics, since a force that varies with position produces a net change in momentum given by the integral of the force. The hypotheses matter more than the statement: continuity of the integrand can be weakened to bounded variation, and the modern formulation in terms of absolutely continuous functions reveals why, since such functions are precisely those that recover their values by integrating their derivative almost everywhere. Improper integrals, taken over unbounded intervals or across a singularity, are declared convergent when a limit of truncated integrals exists, and the comparison test, sometimes supplemented by the Cauchy principal value, decides the standard cases. The harmonic integral diverges, the Gaussian integral converges and evaluates through a rotation of coordinates, and examples such as the conditionally convergent alternating harmonic integral show that convergence of an improper integral need not be robust under rearrangement of the underlying series. Measure and integration generalise all of this. The Riemann integral is defined for functions whose discontinuities form a set small enough to be negligible, a condition that is inconvenient to state precisely and excludes many natural functions, and Lebesgue's insight was to organise the integral around the size of sets rather than around the order of points, replacing upper and lower sums by a supremum over measurable functions bounded above and by an infimum below. This rearrangement costs little and buys a great deal: indicators of measurable sets integrate to their measure, nonnegative functions integrate in the sense that the integral of a sum is the sum of the integrals, the monotone convergence theorem allows limits to be interchanged with integrals without extra assumptions, and dominated convergence does the same under an integrable bound, two results that are proved routinely and used constantly. The measure theoretic forms of Fubini and Tonelli then justify integrating over one variable and the other in any order, which is exactly the property that makes probabilistic independence and the expectation of a bivariate function well behaved. The spaces of measurable functions carrying finite powers of the integral become the Lebesgue spaces, and the inequality relating integrals of products to the corresponding powers of the integrals supplies the only setting in which Holder, Minkowski and the notion of almost everywhere convergence fit together comfortably. For a student who will use probability, statistics, or numerical analysis, this material is not an end in itself but the machinery that explains why expectations behave as they do, why almost sure statements need not hold pointwise, and why a numerical method that converges for almost every input is the strongest guarantee one can usually provide. A brief word on functional analysis completes the survey, and the next paragraph takes it up on its own terms. The reformulation in terms of absolutely continuous functions also repairs a genuine defect in the classical theory. There exist functions that are continuous everywhere and differentiable almost everywhere, with derivative zero almost everywhere, yet which are not constant, and for such functions the second face of the fundamental theorem fails exactly as it does for the Cantor function described earlier. Lebesgue integration restores the identity by adding the qualifier almost everywhere, so that a function equals the integral of its derivative except on a set of measure zero, which is a statement about equivalence classes rather than about pointwise values. In probability this is not a technicality but a matter of routine, since an event may have probability zero while remaining a possible outcome, expectations are defined only up to null sets, and the statements that a random variable converges almost surely or in measure concern subsets of the sample space of vanishing measure. For the engineer the practical lesson is that integrals of measured quantities are insensitive to the exact values taken on null sets, so a model may ignore the behaviour of a field on a set of measure zero without altering any integral derived from it.
