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Chapter 5: Applications of Integration
Section 5.1: Area of a Plane Region
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Example 5.1.4
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Calculate the area of the region bounded by the graphs of , and .
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Solution
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Mathematical Solution
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Integration by vertical strips
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The animation in Figure 5.1.4(a) shows the region shaded in blue, and the graphs of , and in black, red, and green, respectively.
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The dotted vertical line passes through through , the intersection of the graphs of and .
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As the animation progresses, a yellow vertical strip sweeps through that portion of that lies to the left of the dotted line segment. As the strip passes to the rightmost portion of , its color changes to cyan as a reminder that the lower bound of integration changes across .
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The coordinates of point are
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module()
local p1,p2,p3,p4,p5,G,M;
p1:=plot([1,sqrt(x),1-3*x/2],x=0..1,view=[0..1,0..1],labels=[x,y],color=[black,red,green],thickness=[4,3,3]):
p2:=plots:-inequal({y<=1,y>=sqrt(x),y>=1-3*x/2},x=0..1,y=(sqrt(7)-1)/3..1,color=blue,transparency=.8):
p3:=plot([[2*(4-sqrt(7))/9,1],[2*(4-sqrt(7))/9,(sqrt(7)-1)/3]],linestyle=dot,color=black):
p4:=plots:-textplot([.35,.55,typeset(A)]):
p5:=plots:-display(p1,p2,p3,p4):
G:=proc(s)
local S,P;
S:=evalf(2*(4-sqrt(7))/9):
if s<=S then P:=plot([[s,1-3*s/2],[s,1]],color=yellow,thickness=10);
else P:=plot([[s,sqrt(s)],[s,1]],color=cyan,thickness=10);
end if;
end:
M:=plots:-animate(G,[x],x=0..1,frames=31,background=p5,digits=3):
print(M);
end module:
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Figure 5.1.4(a) Area of by vertical strips ()
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The area of is given by
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Integration by horizontal strips
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The animation in Figure 5.1.4(b) shows the region shaded in blue, and the graphs of , and in black, red, and green, respectively.
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The coordinates of point , the intersection of the graphs of and , are
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The animation shows a horizontal strip moving through , from point to the line . Its ends remain on the red and green bounding curves, so the integration by horizontal strips can be accomplished by a single definite integral.
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>
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module()
local p1,p2,p3,p4,p5,G,M;
p1:=plot([1,sqrt(x),1-3*x/2],x=0..1,view=[0..1,0..1],labels=[x,y],color=[black,red,green],thickness=[4,3,3]):
p2:=plots:-inequal({y<=1,y>=sqrt(x),y>=1-3*x/2},x=0..1,y=(sqrt(7)-1)/3..1,color=blue,transparency=.8):
p3:=plots:-textplot([.35,.55,typeset(A)]):
p4:=plots:-display(p1,p2,p3):
G:=s->plot([[2*(1-s)/3,s],[s^2,s]],color=yellow,thickness=10):
M:=plots:-animate(G,[y],y=(sqrt(7)-1)/3..1,frames=21,background=p4,digits=3):
print(M);
end module:
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Figure 5.1.4(b) Area of by horizontal strips ()
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The area of is given by
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≐
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Maple Solution
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Define the functions and
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Control-drag the expression for .
Context Panel: Assign to a Name≻
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Control-drag the expression for .
Context Panel: Assign to a Name≻
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Solve to determine the coordinates of point
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Set and press the Enter key.
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Context Panel: Solve≻Obtain Solutions for≻
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Context Panel: Assign to a Name≻
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Expression palette: Evaluation template
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻
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=
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Integration by vertical strips
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Expression palette: Definite-integral template
Press the Enter key.
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Context Panel: Simplify≻Simplify
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Context Panel: Approximate≻10 (digits)
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Integration by horizontal strips:
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Write the equation and press the Enter key.
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Context Panel: Solve≻Obtain Solutions for≻
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Context Panel: Assign to a Name≻
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Expression palette: Definite-integral template
Press the Enter key.
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Context Panel: Simplify≻Simplify
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Context Panel: Approximate≻10 (digits)
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