Chapter 4: Integration
Section 4.6: Average Value and the Mean Value Theorem
Example 4.6.1
Find the average value of fx=x2 on the interval 0,1.
Solution
Mathematical Solution
The average value, computed as per Definition 4.6.1, is
favg=11−0∫01x2 ⅆx = x3301 = 13
The top of the shaded area in Figure 4.6.1(a) is the horizontal line y=1/3, the average value. The area of the rectangle shaded in the color cyan equals the area bounded by the graph of f, the x-axis, and the line x=1.
The animation in Figure 4.6.1(b) shows the area under f in pink. As the animation slider is advanced, this area "drops down" into a cyan rectangle of area equal to what has "dropped." When all the pink area has dropped, the area of the cyan rectangle will equal the area of the region originally shaded in pink.
Figure 4.6.1(a) Average value of f on 0,1
FXZJ:=proc(s) local p1,p2,p3,p4; p1:=plot(x^2,x=0..1,color=black): p2:=plot(x^2,x=0..1-s,filled=[color=pink]): p3:=plot((1-s)^2,x=1-s..1,filled=[color=pink]): p4:=plot(-2/3-(2/3)*s*sqrt(1-s)+(2/3)*sqrt(1-s)+s,x=0..1,filled=[color=cyan]): plots:-display(p1,p2,p3,p4,scaling=constrained,tickmarks=[2,[1/3=typeset(1/3),2/3=typeset(2/3),1=typeset(1)]]); end proc: plots:-animate(FXZJ,[s],s=0..1,frames=31,paraminfo=false);
Figure 4.6.1(b) Animation: average value of f
Maple Solutions
Solution by tutor
Figure 4.6.1(c) shows the result of applying the tutor to fx=x2 on 0,1. 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 4.6.1(c) 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
11−0 ∫01x2 ⅆx = 13
Table 4.6.1(a) details the use of the FunctionAverage command.
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
FunctionAveragex2,x=0..1,output=integral = ∫01x2ⅆx
FunctionAveragex2,x=0..1 = 13
Table 4.6.1(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 4.6.1(c). If in addition the option averageoptions=color=cyan,filled=true is included, the graph will resemble the one in Figure 4.6.1(a).
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