The symbols 𝑑𝑥, 𝑑𝑦, and 𝑑𝑥/𝑑𝑦 or 𝑑𝑦/𝑑𝑥 were introduced by German mathematician Gottfried Wilhelm Leibniz (1646 – 1716) in a manuscript dated November 11, 1675 (Cajori Vol. 2, p. 204).

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Four classic results from real analysis appear here side by side. The Gaussian ∫₋∞⁺∞ e⁻ˣ² dx = √π The arctangent ∫₋∞⁺∞ 1/(x²+1) dx = π Dirichlet’s ∫₋∞⁺∞ (sin x)/x dx = π And ∫₋∞⁺∞ cos(x)/(1+x²) dx = π/e Digital audio codecs rely on the sinc integral to enforce the Nyquist rate that prevents aliasing in compressed music files.

Beautiful double integral evaluation: ∫₀¹ ∫₀¹ 1 / [(1−xy)(1+x)(1+y)] dx dy = ln 2 The surface plot shows the function f(x,y) over the unit square. Solved using the clever substitution y = u/x, partial fractions, and integration by parts; beautifully simplifying to ln 2.

The curve of e⁻ˣ² spans the real line with total area exactly √π. Square the integral I = ∫₋∞⁺∞ e⁻ˣ² dx to obtain the double integral of e⁻⁽ˣ²⁺ʸ²⁾ over the plane. Switch to polar coordinates (x = r cos θ, y = r sin θ, dx dy = r dr dθ), integrate θ from 0 to 2π and r from 0 to ∞, then substitute u = r² to reduce I² to π. The positive root is therefore √π. This constant normalizes the probability density of the normal distribution that models measurement error in experimental physics.

Gabriel’s Horn Paradox – the infinite trumpet that breaks calculus! Volume (left): ∫ π (1/x)² dx from 1 to ∞ = π (finite) ∫ 1/x dx from 1 to ∞ = ∞ (infinite) One solid you could fill with finite paint but could never paint the outside.