Calculus 2 — Integrals & Series
Riemann sums ground out by the engine vs the exact integral, the fundamental theorem, and series that converge (or don’t).
The integral answers the mirror question: not "how fast?", but "how much, in total?" Distance from speed, energy from power, area from height. The honest definition is brute force — slice thin, add up — and the engine is happy to do it the honest way.
- 01 The Riemann sum — area by brute force Area under f(x) = x² from 0 to 1: slice it into 100 rectangles of width 0.01, add them up. The exact answer (you'll see why below) is 1/3:
- 02 The fundamental theorem Here is the miracle: integration undoes differentiation. To get the area under x², find a function whose derivative IS x² — that's F(x) = x³/3 — and just…
- 03 Series — infinite sums with finite answers Add 1 + ½ + ¼ + ⅛ + ⋯ forever and you do NOT get infinity — each term covers half the remaining distance to 2. A geometric series with ratio |r| < 1…
- 04 Taylor series — polynomials impersonating functions Near a point, any smooth function can be impersonated by a polynomial built from its derivatives. The first two terms of sin(x) are x − x³/6 — plotted below;…
next course: Linear Algebra — Matrices →