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:
\[\int_{-2}^{2} 4 x \,\dd x\]

#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:
\[\begin{array}{rcl}
F(x) &=&\displaystyle \int f(x) \; \dd x \\
&&\phantom{xxx}\blue{\text{definition of the antiderivative}}\\
&=&\displaystyle \int 4 x \; \dd x \\
&&\phantom{xxx}\blue{\text{substituted }f(x)=4 x \text{ into the equation}}\\
&=&4\cdot \displaystyle\int x\,\dd x\\
&&\phantom{xxx}\blue{\text{applied the constant multiple rule: }\displaystyle \int cx^n \; {\dd}x = c\cdot \displaystyle \int x^n\;{\dd}x \text{ with }c=4}\\
&=&4 \left(\displaystyle \cfrac{x^2}{2}+ C\right)\\
&&\displaystyle \phantom{xxx}\blue{\text{applied the reverse power rule:} \int x^{n} \; \dd x = \displaystyle\cfrac{x^{n+1}}{n+1} + C \text{ with }n=1}\\
&=&\displaystyle 2 x^2 + C\\
&&\phantom{xxx}\blue{\text{simplified}}\\
&=&\displaystyle 2 x^2\\
&&\phantom{xxx}\blue{\text{omitted the constant of integration}}\\
\end{array}\]
Now that the antiderivative is known, the definite integral can be calculated:
\[\begin{array}{rcl}
\displaystyle \int_{a}^{b} f(x) \,\dd x&=& F(b) - F(a)\\
&&\phantom{xxx}\blue{\text{definition of a definite integral}}\\
\displaystyle \int_{-2}^{2} 4 x \,\dd x&=&\displaystyle \left(2 (2)^2\right) - \left(2 (-2)^2\right) \\
&&\phantom{xxx}\blue{\text{substituted the boundary values into the antiderivative}}\\
&=&\displaystyle8-8\\
&&\phantom{xxx}\blue{\text{simplified}}\\
&=&\displaystyle 0\\
&&\phantom{xxx}\blue{\text{simplified}}
\end{array}\]

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