Definite integral

Integration: The definite integral

Definite integral

Assume that $\orange F$ is an antiderivate function of the function $\blue f$ . The definite integral of $\blue f$ with lower bound $a$ and upper bound $b$ is defined as:

$\int_a^b \blue f(x) \; \dd x = \orange F(b) - \orange F(a)$

In worked out solutions we often use the notation $\left[\orange F(x)\right]_a^b$ . This is short for $\orange F(b) - \orange F(a)$ .

Example

$\begin{array}{rcl}\displaystyle \int_0^3 \blue{x^2} \; \dd x &=& \left[\orange{\frac{1}{3}x^3}\right]_0^3\\ &=& \frac{1}{3} \cdot 3^3-\frac{1}{3} \cdot 0^3\\ &=& 9-0 \\ &=&9 \end{array}$

Integration constant

Since both $\orange F(b)$ and $\orange F(a)$ have the same integration constant $\green C$ , for the solution it does not matter which value of $\green C$ we pick. Since it's easier to calculate with, we often pick $\green C=0$ .

Rules of calculation

For definite integrals the same rules of calculation hold as for indefinite integrals.

Calculate the following definite integral:
$\int_{-4}^{4} 6 x \,\dd x$

$0$

Definite integrals are calculated with the following formula:
$\displaystyle \int_{a}^{b} f(x) \,\dd x = F(b) - F(a)$
Thus, in order to calculate a definite integral, we first need to determine the antiderivative of the function:
$\begin{array}{rcl} F(x) &=&\displaystyle \int f(x) \; \dd x \\ &&\phantom{xxx}\blue{\text{definition of the antiderivative}}\\ &=&\displaystyle \int 6 x \; \dd x \\ &&\phantom{xxx}\blue{\text{substituted }f(x)=6 x \text{ into the equation}}\\ &=&6\cdot \displaystyle\int x\,\dd x\\ &&\phantom{xxx}\blue{\text{applied the constant multiple rule: }\displaystyle \int cx^n \; {\dd}x = c\cdot \displaystyle \int x^n\;{\dd}x \text{ with }c=6}\\ &=&6 \left(\displaystyle \cfrac{x^2}{2}+ C\right)\\ &&\displaystyle \phantom{xxx}\blue{\text{applied the reverse power rule:} \int x^{n} \; \dd x = \displaystyle\cfrac{x^{n+1}}{n+1} + C \text{ with }n=1}\\ &=&\displaystyle 3 x^2 + C\\ &&\phantom{xxx}\blue{\text{simplified}}\\ &=&\displaystyle 3 x^2\\ &&\phantom{xxx}\blue{\text{omitted the constant of integration}}\\ \end{array}$
Now that the antiderivative is known, the definite integral can be calculated:
$\begin{array}{rcl} \displaystyle \int_{a}^{b} f(x) \,\dd x&=& F(b) - F(a)\\ &&\phantom{xxx}\blue{\text{definition of a definite integral}}\\ \displaystyle \int_{-4}^{4} 6 x \,\dd x&=&\displaystyle \left(3 (4)^2\right) - \left(3 (-4)^2\right) \\ &&\phantom{xxx}\blue{\text{substituted the boundary values into the antiderivative}}\\ &=&\displaystyle48-48\\ &&\phantom{xxx}\blue{\text{simplified}}\\ &=&\displaystyle 0\\ &&\phantom{xxx}\blue{\text{simplified}} \end{array}$

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