∫
dx
The Accumulation Engine
Integration, Area, & The Fundamental Theorem
The Central Question
World 18 asked: How fast is something changing? Now we reverse the question: If I know how something is changing, can I reconstruct the quantity itself? Or, if I know tiny pieces of something, can I add infinitely many of them to recover the whole? This is the birth of Integration.
RATE →
REVERSE RATE →
ANTIDERIVATIVE →
INTEGRAL →
ACCUMULATION →
LIMIT OF SUMS →
DEFINITE INTEGRAL →
AREA →
TOTAL CHANGE →
FUNDAMENTAL THEOREM
The 60 Integral Cells
From Antiderivatives to Global Accumulation
CELL 01-05
Foundations
∫f(x)dx
→
F(x)+C
- The Idea of Accumulation
- Reverse of Differentiation
- Antiderivative
- Why +C (Family of functions)
- Indefinite Integral Notation
CELL 06-11
Basic Integration
∫ xⁿ dx = xⁿ⁺¹/(n+1)
∫ 1/x dx = ln|x|
∫ cos(x) dx = sin(x)
- Power Rule & Constant Multiple
- Sum / Difference
- The Logarithmic Exception (1/x)
- Exponential Integrals
- Basic Trigonometric Integrals
CELL 12-20
Integration Methods
x²
u
- Substitution (Reversing Chain Rule)
- Integration by Parts (∫udv)
- Choosing u (LIATE heuristic)
- Partial Fractions Decomposition
- Linear & Quadratic Factors
CELL 21-26
Standard Integrals
∫
dxx² + a²
→
tan⁻¹
- Forms: 1/(x² ± a²)
- Forms: 1/√(x² ± a²) and 1/√(a²-x²)
- Forms: 1/(ax²+bx+c)
- Related Quadratic & Radical Forms
CELL 27-34
Definite Integrals
∫ba
F(x)
- Riemann-Sum Idea & Rectangles
- The Fundamental Theorem
- Evaluation without +C
- Properties (Reversing, Splitting)
- Substitution with Limits
CELL 35-43
Area Under Curves
- Signed Area vs Geometric Area
- Area Above & Below Axis
- Area Between Two Curves
- Curve Intersections as Limits
- Area of Circles & Ellipses via Calculus
CELL 44-50
Accumulation Applications
- Rate → Total Change
- Velocity → Displacement / Distance
- Density → Mass
- Marginal Cost → Total Cost
- Average Value of a Function
CELL 51-56
Frontiers
∞
- Improper Integrals (Limits to ∞)
- Numerical Integration (Trapezoidal)
- Riemann Sum ↔ Infinite Series
- Horizontal Accumulation (dy)
CELL 57-60
The Accumulation Engine
d/dx
∫
- Integral as Global Accumulation
- Derivative ↔ Integral Duality
- The Fundamental Theorem Bridge
- Mastery of The Accumulation Engine
Exit Portal
We now understand the twin engines of calculus: measuring the instant of change, and gathering those instants into a total accumulation. But what happens when we use equations containing derivatives to describe the physical laws of the universe? It is time to enter the ultimate modeling space.