Chapter 4: Integration
Section 4.6: Average Value and the Mean Value Theorem
Example 4.6.2
Obtain the average value of fx=sinx on the interval 0,π.
Solution
Mathematical Solution
The average value, computed as per Definition 4.6.1, is
favg
=1π−0∫0πsinx ⅆx
=1π−cosx0π
=−1πcosπ−cos0
=−1π−1−1
=2/π≐0.63662
Figure 4.6.2(a) Average value for sinx on 0,π
Figure 4.6.2(a) is a graphical representation of the average value for sinx on the interval 0,π.
Maple Solutions
Solution by tutor
Figure 4.6.2(b) shows the result of applying the tutor to fx=sinx on 0,π. The graph is drawn, the average value calculated, and a dotted line corresponding to the average value is appended to the graph.
The tutor gives simplified access to the FunctionAverage command, which will return either the average value, or a figure comparable to Figure 4.6.1(a).
The Maple command at the bottom of the tutor shows the syntax that generates the graph provided by the tutor.
Figure 4.6.2(b) Function Average tutor
Solution from first principles
Write 1/b−a as a multiplier of Calculus palette: Definite Integral template
Context Panel: Evaluate and Display Inline
1π−0 ∫0πsinx ⅆx = 2π
Table 4.6.2(a) details the use of the FunctionAverage command.
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
FunctionAveragesinx,x=0..π,output=integral = ∫0πsinxⅆxπ
FunctionAveragesinx,x=0..π = 2π
Table 4.6.2(a) Direct use of the FunctionAverage command
The option output=plot is also valid, and leads to a graph similar to the one shown in Figure 4.6.2(b). If in addition the option averageoptions=color=cyan,filled=true is included, the graph will resemble the one in Figure 4.6.2(a).
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