Integral Calculus
Adding up the rectangles…
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Integral Calculus
Adding up the rectangles…
The mathematics of accumulation. Fill an area with rectangles, let them get thin, and the sum becomes an integral — an area, a volume, a total — tied back to derivatives by one theorem.
Fill the area under a curve with rectangles — left, right or midpoint — and watch the error shrink as the rectangles get thinner.
Σ f(xᵢ) ΔxLimitsRectangles Under a CurveThe limit of Riemann sums is a signed area: pieces above the axis count positive, pieces below count negative.
∫ₐᵇ f(x) dx = lim n→∞ Σ f(xᵢ) ΔxRiemann SumsSigned AreaSweep the right edge of an area along the axis and plot how much has accumulated. The slope of that curve at every x is exactly f(x).
d/dx ∫ₐˣ f(t) dt = f(x)The Definite IntegralAccumulationChange the variable and the axis stretches: each thin strip gets wider by u′(x) and shorter by the same factor, so the area survives intact.
∫ f(g(x)) g′(x) dx = ∫ f(u) duFundamental Theorem of CalculusChange of VariableTrace the curve (u, v) and the rectangle under its end point splits into two regions: ∫u dv on one side of the curve and ∫v du on the other.
∫ u dv = uv − ∫ v duDerivative RulesThe Product Rule ReversedFind where two curves meet, then add up thin strips of height top minus bottom — splitting the range wherever the curves swap places.
A = ∫ₐᵇ (top − bottom) dxThe Definite IntegralVertical StripsSpin a region around an axis in 3D. Slice the solid into disks or washers, add up their volumes, and watch the total converge on a cone, a sphere or a paraboloid.
V = ∫ₐᵇ π f(x)² dxArea Between CurvesDisks and WashersTrapezoids, midpoints and Simpson's parabolas side by side. Double n and the errors fall by 4, 4 and 16 — straight lines on a log scale.
Simpson: (Δx/3)(f₀ + 4f₁ + 2f₂ + … + fₙ)Riemann SumsError OrdersDistance from speed, charge from current, work from force: whenever you know how fast something accumulates, the integral gives the total.
Every continuous probability is an area under a density curve, and every expected value is an integral. Statistics is integration in disguise.
Most integrals have no closed form. Physics engines, finance models and ML all rely on the trapezoid and Simpson rules — and on knowing how fast their errors shrink.