11. Integration

From the Riemann integral to the Lebesgue integral — how the idea of integration expands.

• The Gamma Function — The factorial stretched to real arguments. Values that jumped like stairs become one smooth curve. The key rule: shifting by one multiplies by the shift amount. That rule fills in the values between integers. An integral, an area, draws the whole curve. For example, picture a 🪜 staircase jumping 1, 2, 6, 24, with a smooth slide fitted between its steps. Now the heights between steps can be measured too.
• The Beta Function — The area under a curve shaped by two exponents, p and q, is the beta function. Sliders morph the curve's shape. The two ends of the curve are pulled with different strengths. So the hump shifts and changes height. This area equals an exact ratio of gamma functions. For example, pull the two ends of a 🧵 string with different strengths and the bulge in the middle changes shape. Adjusting the end pulls is what p and q do.
• Bohr–Mollerup Theorem — Two rules alone do not pin down the factorial's extension. An impostor matching the gamma function at every integer survives. One extra condition filters the impostor out: log-convexity. The graph of the logarithm must be convex. The secant test exposes this fingerprint. For example, height and weight alone cannot confirm a 🧑 friend's identity. A matching fingerprint settles it. Log-convexity is the gamma function's fingerprint.
• Improper Integrals — Integrals over intervals that run forever or blow up. The exponent p decides convergence against divergence. An endless region can still have finite area. Its tail just has to thin out fast enough. Too thick a tail piles area up forever. For example, a 💧 water leak that slows down quickly spills only one bucketful, even leaking forever. A leak that slows too little overflows the bucket in the end.
• Gabriel's Horn — Spin a curve into a horn and meet a paradox. The volume is finite but the surface area is infinite. Filling the inside and painting the outside behave differently. The thinning rate splits the two computations. One total stops growing while the other never does. For example, a coiled 🧵 string fits inside a small 📦 box. Spread that same string flat across the floor and it would need an endless room.
• Upper and Lower Sums Close In — Rectangles pack the region under a curve to measure its area. Lower and upper sums squeeze the true area from both sides. The answer sits between the underfilled and overfilled totals. Finer partitions narrow the gap between them. When the gap vanishes, that value is the integral. For example, fit square pieces into the curved shape of a 🍰 cake. The smaller you cut the pieces, the fewer gaps remain and the better they match.
• The Dirichlet Function — A function that is 1 on the rationals and 0 on the irrationals. It is the canonical failure of the Riemann integral. Every tiny interval holds rationals and irrationals together. So the upper sum stays pinned at 1. The lower sum stays at 0 and the gap never closes. For example, mix black and white ⚽ balls as finely as you like, and every scoop still holds both colors. No single height can be assigned.
• Volterra's Function — Differentiable everywhere with a bounded derivative, yet that derivative is not Riemann integrable. Bumps rise over a fat Cantor set. The function itself joins up smoothly. The trouble is its derivative, oscillating hard on a thick set. Riemann sums cannot capture that oscillation. For example, measure the slope of a smooth road with a 📏 ruler, but the reading jumps wherever the ruler touches. The road is fine; the measurement fails.
• The Area Function's Derivative Is the Function — Measure the slope of the accumulated-area function. It matches the original function exactly. Area grows at a rate equal to the height at that point. Nudge the right edge and a thin strip attaches. The strip's area is height times the nudge. For example, filling a tank with 💧 water, the level rises at the rate the water pours in. Differentiate what has accumulated and you recover what is pouring.
• Integration by Parts — Run the product rule backwards to trade one integral for another. The area splits into two pieces, each computed separately. A hard integral becomes a boundary term plus an easier integral. In the area picture, one big rectangle splits into two pieces. Both computations agree to the last decimal. For example, measuring a 🍰 cake's area gets easier when you swap to a full rectangle minus one piece. Split the measuring and even awkward shapes become manageable.
• Change of Variables — Stretch and squeeze the axis to rewrite an integral. Two different substitutions land on the same value. Change the scale and the endpoints move along. Areas need correcting by the stretch factor. After the correction, a hard integral turns easy. For example, measure with a 📏 ruler whose marks are stretched twofold and every reading doubles. Halving the reading corrects it back to the true length.
• Functions of Bounded Variation — This scene measures the total distance a function wiggles. If that sum stays under a finite ceiling, the function has bounded variation. Start and finish alone are not enough. Every up and down in between must be added. Monotone functions pass since they wiggle one way only. Endless trembling makes the sum explode. For example, a 🚗 car driving up and down has bounded variation while its total mileage stays definite. Trembling in place forever would pile the distance up without end.
• Arc Length — This scene measures a curve by cutting it into straight pieces. The sum of the piece lengths is the polygonal length. Finer partitions only make that sum grow, approaching the true length from below. The supremum over all partitions is the length of the curve. If that supremum is infinite, the curve cannot be measured. For example, lay a 🧵 string along a winding path and measure it piece by piece, straightening each bit. The finer the pieces, the closer you get to the string's true length.
• The Riemann–Stieltjes Integral — An integral that weighs intervals by the integrator's growth, not their length. Change the integrator and the same intervals count differently. Each interval can carry a different weight. A step integrator dumps all its weight on one point. Then the integral is decided by that point alone. For example, total cargo by weight, not by count, when each 📦 box weighs differently. If one box holds all the weight, it alone decides the total.
• Jordan Decomposition — Every bounded-variation function splits into a difference of two increasing functions. Drag the probe and watch the identity hold pointwise. Record the total rise and the total fall separately. Subtract falls from rises to get the current height. The variation function is the running record of rises. For example, money now in a 🐷 piggy bank is the total 💰 money put in minus the total taken out. Rises and falls are counted on separate tallies.
• Lebesgue Measure — The cheapest interval cover prices a set's size. A single point, and even all the rationals, cost nothing. Length measurement extends to sets like points and rationals. Among all covers, take the smallest total length. That smallest total is the set's size. For example, cover a set with pieces of 🧵 string; the shortest total string length is the set's size. A single point needs almost no string, so its size is zero.
• The Vitali Set — Pick one representative from each equivalence class using the axiom of choice. The result is a set whose size cannot be measured. Shifted copies of this set cover a whole interval. If its size were zero, the copies would total zero. If positive, the copies would total infinity. Both contradict, so no size exists. For example, copies of one 📦 box can fill a room, yet if the box's size were zero the total should be zero, and if it were anything more the total would explode. So the size cannot be assigned.
• Borel–Cantelli Lemmas — If total length is finite, points hit infinitely often form a null set. This scene checks that boundary numerically. Collect the points where the events keep overlapping. Finite total length makes the overlaps rarer and rarer. With independence, the converse also holds. For example, in rain whose 💧 water drops keep shrinking, a finite total volume means one spot is almost never hit endlessly.
• Covering the Rationals with Nothing — Cover the rationals one by one and the total length drops below any bound. The Riemann integral cannot move here. The rationals are countably many. So they can be numbered and given shrinking covers. The covers' total can be made as small as you wish. For example, cover 🍬 candies scattered on the floor, one by one, with smaller and smaller cups, and the cups' total size shrinks as low as you like.
• Lebesgue Convergence Theorems — Three theorems license swapping limits and integrals. Examples with escaping mass show why each hypothesis is needed. Swapping the order requires that no mass leaks away. A dominating function overhead keeps things safe. Without conditions, area can vanish in the limit. For example, pour 💧 water from cup to cup with no spill and the total stays the same. One drop escaping sideways shrinks the total that arrives.
• Fubini–Tonelli Theorem — Iterated integrals agree in either order. Sum the rows first or the columns first, the total is the same. For nonnegative functions this is always safe. For integrable functions it is safe too. Without integrability, the two orders can split apart. For example, count 🍎 apples packed in a grid inside a 📦 box: counting by rows or by columns gives the same total.
• Convolution — Flip one function, slide it, and measure how it overlaps the other. Overlapping two indicator blocks gives an overlap length that traces a tent curve. Convolving with a smooth kernel softens even a sharp step function. The narrower the kernel, the closer the softened function sits to the original. For example, it is the blur effect in a 📷 photo, replacing each pixel by a weighted average of its neighbors. The center pixel gets the largest weight.
• Egorov's Theorem — On a finite-measure set, pointwise convergence is almost uniform. Squeeze the bad set small and the rest converges uniformly. The slow points hide inside one small set. Remove that set and the rest converges at one pace. The whole space must have finite measure. For example, of 30 🧑 friends racing to a finish line, one or two may lag; set them aside and the rest arrive at nearly the same pace.
• Absolute Continuity — Shrink the total length of intervals to zero and the total change shrinks too. The Cantor function is a continuous function failing this. Tiny input changes must not create large outputs. Such functions are exactly the ones built as integrals. The Cantor function stays flat yet climbs from 0 to 1, breaking the rule. For example, open a tap a little and only a little 💧 water comes out. The Cantor function is a strange tap that pours a bucketful while barely opened.
• Differentiation Under the Integral Sign — Numerics confirm that differentiating inside the integral matches the tangent slope. Drag the point and its tangent along the curve. Differentiating the whole or the inside gives the same answer. Each point's change accumulates into the total change. A domination condition makes the swap safe. For example, measure how fast a room's average 🌡️ thermometer reading rises: averaging how fast each spot warms up gives the same answer.
• Minkowski's Inequality — The size of a sum of two functions never exceeds the sum of their sizes. Slide the exponent p and the gap between the bars opens and closes. This is the triangle inequality moved to functions. A straight route is never longer than two joined legs. At exponent 1 the gap closes completely. For example, a 🚗 car that drives two legs in an L shape travels at least the straight-line distance, never more than the two legs added together.