Chapter 5: Applications of Integration
Section 5.1: Area of a Plane Region
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Example 5.1.3
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Calculate the area bounded by the graphs of and .
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Solution
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Mathematical Solution
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According to Table 5.1.1, the area between and is given by the definite integral
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The animation in Figure 5.1.3(a) illustrates why this is so. In frame 1 of the animation, is graphed in black; , in green; and the shaded area is the area "under " given by the integral of . The area under , the green curve, given by the integral of , must be subtracted from the area under . Hence, the difference
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module()
local F,p1,p2,p3,S,SS,G;
F:=plottools:-transform(unapply([x,y+x^2],(x,y))):
p1:=plot(sqrt(x)-x^2,x=0..1,y=0..1,filled=[color=orange,transparency=.5]):
p2:=plot([sqrt(x),x^2],x=0..1,y=0..1,thickness=2,color=[black,green]):
p3:=plots:-display(F(p1),p2):
G:=s->plot(x^2,x=0..1,filled=[color=orange,transparency=s],color=green,thickness=2):
SS:=plots:-animate(G,[t],t=.5..1,background=p3,paraminfo=false):
print(SS);
end module:
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Figure 5.1.3(a) Animation: Area between curves
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As the animation in Figure 5.1.3(a) progresses, the subtraction of the area under from the area under is depicted by the gradual fading of the shading under the graph of , so that in frame 25 the only shading remaining is that between the graphs of and .
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Maple Solutions
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The region between two curves can be shaded with the task template in Table 5.1.3(a).
Tools≻Tasks≻Browse: Algebra≻Shading between Curves
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Line thickness:
Transparency:
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Table 5.1.3(a) Shading the region between two curves
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Calculation of the area between two curves can be done in a syntax-free mode, as shown in Table 5.1.3(b).
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Expression palette: Definite-integral template
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Context Panel: Evaluate and Display Inline
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Table 5.1.3(b) Calculation of the area between and
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The animation in Figure 5.1.3(b) shows a vertical strip of width whose horizontal sweep from to covers the complete region between the graphs of and . This corresponds to the integration in Table 5.1.3(b).
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module()
local p1,G,S;
p1:=plot([sqrt(x),x^2],x=0..1,color=[black,green],thickness=2):
G:=s->plot([[s,s^2],[s,sqrt(s)]],color=red,thickness=10):
S:=plots:-animate(G,[x],x=0..1,frames=51,background=p1,digits=2):
print(S);
end module:
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Figure 5.1.3(b) Integrate with a vertical strip ()
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module()
local p1,F,S;
p1:=plot([sqrt(x),x^2],x=0..1,color=[black,green],thickness=2):
F:=s->plot([[s^2,s],[sqrt(s),s]],color=red,thickness=10):
S:=plots:-animate(F,[y],y=0..1,frames=51,background=p1,digits=2):
print(S);
end module:
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Figure 5.1.3(c) Integrate with a horizontal strip ()
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The animation in Figure 5.1.3(c) shows a horizontal strip of thickness whose vertical sweep from to covers the complete region between the graphs of and . This corresponds to the integration in Table 5.1.3(c).
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Expression palette: Definite-integral template
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Context Panel: Evaluate and Display Inline
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Table 5.1.3(c) Calculation of the area between and
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For integration with a horizontal strip between the curves and , the inequality must hold. If it does, then think of the functions as and , where defines the leftmost boundary; and , the rightmost. Accordingly, the area by horizontal strips is
where is the smallest -value for the strip and is the largest.
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