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Chapter 4: Integration
Section 4.6: Average Value and the Mean Value Theorem
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Example 4.6.1
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Find the average value of on the interval .
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Solution
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Mathematical Solution
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= =
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The top of the shaded area in Figure 4.6.1(a) is the horizontal line , the average value. The area of the rectangle shaded in the color cyan equals the area bounded by the graph of , the -axis, and the line .
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The animation in Figure 4.6.1(b) shows the area under 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.
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Figure 4.6.1(a) Average value of on
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>
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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);
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Figure 4.6.1(b) Animation: average value of
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Maple Solutions
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Solution by tutor
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Figure 4.6.1(c) shows the result of applying the
tutor to on . The graph is drawn, the average value calculated, and a dotted line corresponding to the average value is appended to the graph.
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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).
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The Maple command at the bottom of the tutor shows the syntax that generates the graph provided by the tutor.
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Figure 4.6.1(c) Function Average tutor
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Solution from first principles
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Write as a multiplier of
Calculus palette: Definite Integral template
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Context Panel: Evaluate and Display Inline
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=
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Table 4.6.1(a) details the use of the FunctionAverage command.
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Tools≻Load Package: Student Calculus 1
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Loading Student:-Calculus1
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=
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=
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Table 4.6.1(a) Direct use of the FunctionAverage command
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The option is also valid, and leads to a graph similar to the one shown in Figure 4.6.1(c). If in addition the option is included, the graph will resemble the one in Figure 4.6.1(a).
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<< Previous Section Section 4.6
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