CalcNotes The Definite Integral

Section 5.3 The Definite Integral

Definition 5.3.1. Definite integral.

Let \(f\) be a function defined on a closed interval \([a,b]\text{.}\) The definite integral of \(f\) over \([a,b]\) is written \(\displaystyle\int_a^b f(x)\, dx\) and is defined by

\begin{equation*} \int_a^b f(x)\, dx = \lim_{||P||\rightarrow 0} \sum f(c_k) \Delta x_k \end{equation*}

provided this number exists, in which case we say \(f\) is integrable over \([a,b]\text{.}\)

Fact 5.3.2.

If \(f\) is continuous on \([a,b]\text{,}\) or if \(f\) has finitely many jump discontinuities, then \(f\) is integrable and \(\displaystyle\int_a^b f(x)\, dx\) exists.

Fact 5.3.3. Properties of the definite integral.

Assuming that \(f\) and \(g\) are integrable functions over \([a,b]\text{:}\)

Order of integration

\(\displaystyle \displaystyle\int_b^a f(x)\, dx = -\int_a^b f(x)\, dx\)

Zero-width

\(\displaystyle \displaystyle\int_a^a f(x)\, dx = 0\)

Constant multiple

\(\displaystyle\int_a^b k(x)\, dx = k\int_a^b f(x)\, dx\) for a constant \(k\)

Sum and difference

\(\displaystyle \displaystyle\int_a^b f(x) \pm g(x)\, dx = \int_a^b f(x)\, dx \pm \int_a^b g(x)\, dx\)

Additivity

\(\displaystyle\int_a^b f(x)\, dx + \int_b^c f(x)\, dx = \int_a^c f(x)\, dx\text{,}\) provided all three exist

Max-min

\(\displaystyle \displaystyle\min(f)\cdot (b-a) \leq \int_a^b f(x)\, dx \leq \max(f) \cdot (b-a)\)

Domination

if \(f(x)\geq g(x)\) at every \(x\) in \([a,b]\text{,}\) then

\begin{equation*} \displaystyle\int_a^b f(x)\, dx \geq \int_a^b g(x)\, dx \end{equation*}

and consequently, if \(f(x) \geq 0\) on \([a,b]\text{,}\) then

\begin{equation*} \displaystyle\int_a^b f(x)\, dx \geq 0 \end{equation*}

Example 5.3.4.

Suppose \(f\) and \(g\) are integrable and

\begin{align} \displaystyle\int_1^2 f(x)\, dx \amp= -4\tag{5.3.1}\\ \displaystyle\int_1^5 f(x)\, dx \amp= 6\tag{5.3.2}\\ \displaystyle\int_1^5 g(x)\, dx \amp= 8\tag{5.3.3} \end{align}

Find the following:

\(\displaystyle \displaystyle\int_2^2 g(x)\, dx\)
\(\displaystyle \displaystyle\int_5^1 g(x)\, dx\)
\(\displaystyle \displaystyle\int_1^2 3f(x)\, dx\)
\(\displaystyle \displaystyle\int_2^5 f(x)\, dx\)
\(\displaystyle \displaystyle\int_1^5 4f(x) - g(x)\, dx\)

Answer.

\(\displaystyle 0\)
\(\displaystyle -8\)
\(\displaystyle -12\)
\(\displaystyle 10\)
\(\displaystyle 16\)

Definition 5.3.5. Area under the curve.

If \(y=f(x)\) is a nonnegative and integrable function over \([a,b]\text{,}\) then the area under the curve \(y=f(x)\) over \([a,b]\) is defined to be

\begin{equation*} A=\int_a^b f(x)\, dx \end{equation*}

Note: negative function means negative area.

Definition 5.3.6. Average over an interval.

If \(f\) is integrable over \([a,b]\text{,}\) then its average value on \([a,b]\text{,}\) also called the mean, is

\begin{equation*} \operatorname{av}(f) = \frac{1}{b-a}\int_a^b f(x)\, dx \end{equation*}

Recall the mean value theorem: \(\displaystyle f'(c) = \frac{f(b)-f(a)}{b-a}\)

Fact 5.3.7. Mean Value Theorem for the definite integral.

If \(f\) is continuous on \([a,b]\text{,}\) then there is a point \(c\) in \([a,b]\) such that

\begin{equation*} f(c) = \frac{1}{b-a}\int_a^b f(x)\, dx \end{equation*}