# 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 $$[a,b]$$.
$$\int_a^b F'(x) \; dx = F(b) – F(a).$$