NapkinCalc

Calculus 2A — Integrals

The Fundamental Theorem of Calculus

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

ELI5: the miracle that names the subject — integration undoes differentiation. Instead of a thousand rectangles, find a function F whose derivative is the integrand (an antiderivative) and just subtract its values at the ends: area = F(1) − F(0). For x², F(x) = x³/3.

F(x)=x33F\left(x\right) = \frac{x^{3}}{3} an antiderivative of x² (since F′ = x²)
Aexact=F(1)−F(0)A_{exact} = \mathrm{F}\left(1\right) - \mathrm{F}\left(0\right) = 0.33330.3333 the whole area in one subtraction: 1/3
✓ pass abs(R1000−Aexact)<0.001\mathrm{abs}\left(R_{1000} - A_{exact}\right) < 0.001 brute force and the theorem agree

The engine integrates symbolically, and (on Pro) the Steps button walks through the antiderivative rule by rule:

integrate(2⋅x,x)\mathrm{integrate}\left(2 \cdot x, x\right) = x2x^{2} antiderivative of 2x is x² — click Steps
integrate(x2,x)\mathrm{integrate}\left(x^{2}, x\right) = x33\frac{x^{3}}{3} antiderivative of x² is x³/3 — Steps explains it

Try it yourself: find the area under 3x² from 0 to 1. (An antiderivative of 3x² is x³, so it's 1³ − 0³.)

areayou=1area_{you} = 1 ✏️ Your turn: compute ∫₀¹ 3x² dx. The check compares your value to 1000 thin rectangles — so you must actually evaluate the integral.
✓ pass abs(areayou−Sum("3∗(i/1000)2∗0.001","i",1,1000))<0.01abs(area_{you} - Sum("3*(i/1000)^2 * 0.001", "i", 1, 1000)) < 0.01 green when your area matches the rectangles

Where next: Calculus 2B runs sums to infinity — series and the Taylor polynomials behind your calculator.