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.

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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