### 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}#

#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 7 x \; \dd x \\

&&\phantom{xxx}\blue{\text{substituted }f(x)=7 x \text{ into the equation}}\\

&=&7\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=7}\\

&=&7 \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 \frac{7}{2} x^2 + C\\

&&\phantom{xxx}\blue{\text{simplified}}\\

&=&\displaystyle \frac{7}{2} 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_{-5}^{5} 7 x \,\dd x&=&\displaystyle \left(\frac{7}{2} (5)^2\right) - \left(\frac{7}{2} (-5)^2\right) \\

&&\phantom{xxx}\blue{\text{substituted the boundary values into the antiderivative}}\\

&=&\displaystyle{{175}\over{2}}-{{175}\over{2}}\\

&&\phantom{xxx}\blue{\text{simplified}}\\

&=&\displaystyle 0\\

&&\phantom{xxx}\blue{\text{simplified}}

\end{array}\]

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 7 x \; \dd x \\

&&\phantom{xxx}\blue{\text{substituted }f(x)=7 x \text{ into the equation}}\\

&=&7\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=7}\\

&=&7 \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 \frac{7}{2} x^2 + C\\

&&\phantom{xxx}\blue{\text{simplified}}\\

&=&\displaystyle \frac{7}{2} 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_{-5}^{5} 7 x \,\dd x&=&\displaystyle \left(\frac{7}{2} (5)^2\right) - \left(\frac{7}{2} (-5)^2\right) \\

&&\phantom{xxx}\blue{\text{substituted the boundary values into the antiderivative}}\\

&=&\displaystyle{{175}\over{2}}-{{175}\over{2}}\\

&&\phantom{xxx}\blue{\text{simplified}}\\

&=&\displaystyle 0\\

&&\phantom{xxx}\blue{\text{simplified}}

\end{array}\]

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