Free lesson · Integral calculus

Antiderivatives and the fundamental theorem

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Start with the idea

Differentiation extracts a rate from an accumulated quantity.
Integration reconstructs the accumulated change from that rate.
The fundamental theorem connects these operations under continuity assumptions.

Symbols, units & horizon

F: an antiderivative of f
F′: its derivative
a,b: fixed bounds
A(x): accumulated integral up to x
t: dummy integration variable
C: arbitrary additive constant, not a closing price
F(b)−F(a): ending accumulated value minus starting value
d/dx: derivative with respect to the upper-bound variable

When and why to use this

Use antiderivatives for exact integrals when available, and use initial conditions to recover absolute levels from rates. Always differentiate a proposed antiderivative to check it.

An antiderivative F of f satisfies F′=f.
Any added constant has zero derivative, so an indefinite integral is a family F(x)+C.
A definite integral returns a number for fixed bounds; the arbitrary constant cancels.
For continuous f, A(x)=∫ from a to x of f(t)dt has derivative f(x).
This explains why evaluating an antiderivative at two endpoints produces the accumulated change.
The integration variable t is a dummy label and differs from the moving upper endpoint x.

F^{'} = f \Rightarrow \int_{a}^{b} f (x) d x = F (b) - F (a), \frac{d}{d x} \int_{a}^{x} f (t) d t = f (x)

Calculus: derivation and arithmetic

Antiderivatives and the fundamental theorem

First reverse the power rule: $f (x) = 3 x^{2}$ has antiderivative $F (x) = x^{3} + C$ , since $F^{'} (x) = 3 x^{2}$ .
Evaluate both endpoints and subtract: $\int_{1}^{2} 3 x^{2} d x = F (2) - F (1) = (8 + C) - (1 + C) = 7$ . The arbitrary constant cancels.
Define accumulated area $A (x) = \int_{a}^{x} f (t) d t$ . Additivity gives $A (x + h) - A (x) = \int_{x}^{x + h} f (t) d t$ .
Divide by the nonzero interval width: $\frac{A (x + h) - A (x)}{h} = \frac{1}{h} \int_{x}^{x + h} f (t) d t$ . This is the local average of the rate.
For continuous $f$ , let $h \to 0$ : the local average tends to $f (x)$ , so $A^{'} (x) = f (x)$ . This limit is the calculus step; endpoint subtraction above is algebra.

Work it by hand

If velocity is v(t)=3t² metres/second with t in seconds and the coefficient carrying the required units, displacement from t=1 to t=2 is 7 metres. An initial position of 100 metres makes the final position 107, not 7.

Architecture pseudocode

def integrate_antiderivative(antiderivative,a,b):
    return antiderivative(b)-antiderivative(a)

def recover_level(initial_level,antiderivative,start,end):
    return initial_level+integrate_antiderivative(antiderivative,start,end)

print(integrate_antiderivative(lambda x:x**3,1,2),recover_level(100,lambda t:t**3,1,2))

An analogy to remember

The slope of an odometer reading is speed. Integrating the speed reconstructs distance travelled, while the starting odometer reading supplies the missing constant.

How this becomes a building block

The theorem is a bridge between rate equations and accumulated values. For deterministic continuous growth, integrating the log-growth rate gives a log wealth factor, which must then be exponentiated. It is also used to evaluate density integrals and continuous cash-flow formulas.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def integrate_antiderivative(antiderivative,a,b):
    return antiderivative(b)-antiderivative(a)

def recover_level(initial_level,antiderivative,start,end):
    return initial_level+integrate_antiderivative(antiderivative,start,end)

print(integrate_antiderivative(lambda x:x**3,1,2),recover_level(100,lambda t:t**3,1,2))

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