5.4 | The Fundamental Theorem of Calculus


The Fundamental Theorem of Calculus
If a function \(f\) is continuous on the closed interval \([a,b]\) and \(F\) is an antiderivative of \(f\) on the interval \([a,b]\), then
$$
\int_a^b f(x) \; dx = F(b) – F(a)
$$
Guidelines for Using the Fundamental Theorem of Calculus

  1. Provided that you can find an antiderivative of \(f\), you now have a way to evaluate the definite integral without having to use the limit of a sum.
  2. When applying the Fundamental Theorem of Calculus, the notation shown below is convenient.
    $$
    \int_a^b f(x) \; dx = F(x) \Big|_a^b = F(b)-F(a)
    $$
    For instance, to evaluate \(\int_1^3 x^3 \; dx\), you can write
    $$
    \int_1^3 x^3 \; dx = \frac{x^4}{4} \Big|_1^3 = \frac{3^4}{4} – \frac{1^4}{4} = \frac{81}{4} – \frac{1}{4} = 20.
    $$
  3. It is not necessary to include a constant of integration \(C\) in the antiderivative.
    $$
    \int_a^b f(x) \; dx = \left( F(x) +C \right)\Big|_a^b = [F(b) + C] -[F(b) + C] = F(b) – F(a).
    $$
Mean Value Theorem for Integrals
If \(f\) is continuous on the closed interval \([a,b]\), then there exists a number \(c\) in the closed interval \([a,b]\) such that
$$
\int_a^b f(x) \; dx = f(c) (b-a).
$$
Definition of the Average Value of a Function on an Interval
If \(f\) is integrable on the closed interval \([a,b]\), then the average value of \(f\) on the interval is
$$
\frac{1}{b-a} \int_a^b f(x) \; dx.
$$
The Second Fundamental Theorem of Calculus
If \(f\) is continuous on an open interval \(I\) containing \(a\), then, for every \(x\) in the interval
$$
\frac{d}{dx}\left[ \int_a^x f(t) \; dt \right] = f(x).
$$
The Net Change Theorem
The definite integral of the rate of change of quantity \(F'(x)\) gives the total change, or net change, in that quantity on the interval \([a,b]\).
$$
\int_a^b F'(x) \; dx = F(b) – F(a).
$$

E 5.4 Exercises