Definite Integration and the Fundamental Theorem of Integral Calculus

So far, every integral in this chapter has been an INDEFINITE integral — a family of functions, ending in $+C$. A definite integral instead has two limits attached and evaluates to a single NUMBER, with no constant of integration left in the final answer.

Notation

The definite integral of $f(x)$ from $x=a$ to $x=b$ is written

where $a$ is the lower limit and $b$ is the upper limit.

The Fundamental Theorem of Integral Calculus (statement only)

The quantity $F(b)-F(a)$ is often written with the shorthand notation $\big[F(x)\big]_{a}^{b}$.

Why the constant of integration disappears

If $F(x)+C$ is used instead of $F(x)$, the constant cancels automatically:

This is exactly why $C$ is never written when evaluating a definite integral — it would cancel out regardless of its value, so any one antiderivative may be used.

Evaluating a definite integral — the three-step method

1. Find an antiderivative $F(x)$ of the integrand, using the standard formulas of this chapter.
2. Evaluate $F(x)$ at the upper limit and at the lower limit.
3. Subtract: $F(\text{upper}) - F(\text{lower})$.

$\int_{1}^{2}(3x^{2}+2x)\,dx = \Big[x^{3}+x^{2}\Big]_{1}^{2} = (2^{3}+2^{2})-(1^{3}+1^{2}) = 12-2 = 10$ …