This section covers the Fundamental Theorem of Calculus for definite integration, the limits of integration and why the constant of integration is omitted.
Definite integration and the theorem
For definite integration, the Fundamental Theorem of Calculus links differentiation and integration over a fixed interval. If f′(x) is integrated from a to b, then
∫abf′(x)dx=f(b)−f(a).
The numbers written beside the integral sign are the limits of integration: a is the lower limit and b is the upper limit.
Notation
Meaning
∫ab
Integrate between the lower limit a and upper limit b.
f′(x)
The function being integrated in this form of the theorem.
f(b)−f(a)
Value at the upper limit minus value at the lower limit.
Why there is no +C
An indefinite integral needs a constant of integration, but a definite integral does not. If the same constant were attached to both endpoint values, it would cancel:
[f(b)+C]−[f(a)+C]=f(b)−f(a).
So a definite integral is evaluated without writing +C.
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