Chapter 5: Applications of Integration
Section 5.1: Area of a Plane Region
Example 5.1.4
Calculate the area of the region R bounded by the graphs of fx=1,gx=1−3 x/2, and hx=x.
Solution
Mathematical Solution
Integration by vertical strips
The animation in Figure 5.1.4(a) shows the region R shaded in blue, and the graphs of f,g, and h in black, red, and green, respectively.
The dotted vertical line passes through through A, the intersection of the graphs of g and h.
As the animation progresses, a yellow vertical strip sweeps through that portion of R that lies to the left of the dotted line segment. As the strip passes to the rightmost portion of R, its color changes to cyan as a reminder that the lower bound of integration changes across A.
The coordinates of point A are 24−7/9,7−1/3≐0.3,0.5
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:
Figure 5.1.4(a) Area of R by vertical strips (dx)
The area of R is given by
∫02 4−7/91−1−3 x/2 ⅆx+∫2 4−7/911−x ⅆx
=−2081+14817
≐0.2103767699
Integration by horizontal strips
The animation in Figure 5.1.4(b) shows the region R shaded in blue, and the graphs of f,g, and h in black, red, and green, respectively.
The coordinates of point A, the intersection of the graphs of g and h, are 24−7/9,7−1/3≐0.3,0.5
The animation shows a horizontal strip moving through R, from point A to the line y=1. Its ends remain on the red and green bounding curves, so the integration by horizontal strips can be accomplished by a single definite integral.
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:
Figure 5.1.4(b) Area of R by horizontal strips (dy)
∫7−1/31y2−2 1−y/3 ⅆy=−2081+14817 ≐ 0.2103767699
Maple Solution
Define the functions g and h
Control-drag the expression for g. Context Panel: Assign to a Name≻g
1−3 x/2→assign to a nameg
Control-drag the expression for h. Context Panel: Assign to a Name≻h
x→assign to a nameh
Solve g=h to determine the coordinates of point A
Set g=h and press the Enter key.
Context Panel: Solve≻Obtain Solutions for≻x
Context Panel: Assign to a Name≻Ax
g=h
1−32x=x
→solutions for x
89−297
→assign to a name
Ax
Expression palette: Evaluation template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻Ay
gx=a|f(x)x=Ax = −13+137→assign to a nameAy
Expression palette: Definite-integral template Press the Enter key.
Context Panel: Simplify≻Simplify
Context Panel: Approximate≻10 (digits)
∫0Ax1−g ⅆx+∫Ax11−h ⅆx
3489−2972−8981+38817
= simplify
−2081+14817
→at 10 digits
0.2103767699
Integration by horizontal strips:
xR=h−1y=y2
xL=g−1y=Gy
Write the equation y=g and press the Enter key.
Context Panel: Assign to a Name≻G
y=g
y=1−32x
23−23y
G
∫Ay1y2−G ⅆy
−29−13−13+1373+297−13−13+1372
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