Calculus 2 — Integrals & Series

The fundamental theorem

continues from lesson 1 — values defined earlier in the course stay live here

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 evaluate it at the ends: F(1) − F(0). No rectangles.

F\left(x\right) = \frac{x^{3}}{3}

an antiderivative of x²

A_{exact} = \mathrm{F}\left(1\right) - \mathrm{F}\left(0\right)

0.3333

the whole area in one subtraction

✓ pass

\mathrm{abs}\left(R_{1000} - A_{exact}\right) < 0.001

brute force and the theorem agree

That is why integrals are computable at all: a thousand rectangle additions collapse into one subtraction.