Chapter 5: Applications of Integration
Section 5.1: Area of a Plane Region
Example 5.1.3
Calculate the area bounded by the graphs of fx=x and gx=x2.
Solution
Mathematical Solution
According to Table 5.1.1, the area between fx=x and gx=x2 is given by the definite integral
∫01fx−gx ⅆx=∫01x−x2 ⅆx = 13
The animation in Figure 5.1.3(a) illustrates why this is so. In frame 1 of the animation, f is graphed in black; g, in green; and the shaded area is the area "under f " given by the integral of f. The area under g, the green curve, given by the integral of g, must be subtracted from the area under f. Hence, the difference
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:
Figure 5.1.3(a) Animation: Area between curves
∫01fx ⅆx−∫01gx ⅆx=∫01fx−gx ⅆx
As the animation in Figure 5.1.3(a) progresses, the subtraction of the area under g from the area under f is depicted by the gradual fading of the shading under the graph of g, so that in frame 25 the only shading remaining is that between the graphs of f and g.
Maple Solutions
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
f1x=
f2x=
≤x≤
≤y≤
Line thickness:
Transparency: =
Table 5.1.3(a) Shading the region between two curves
Calculation of the area between two curves can be done in a syntax-free mode, as shown in Table 5.1.3(b).
Expression palette: Definite-integral template
Context Panel: Evaluate and Display Inline
∫01x−x2 ⅆx = 13
Table 5.1.3(b) Calculation of the area between fx=x and gx=x2
The animation in Figure 5.1.3(b) shows a vertical strip of width dx whose horizontal sweep from x=0 to x=1 covers the complete region between the graphs of f and g. This corresponds to the integration in Table 5.1.3(b).
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:
Figure 5.1.3(b) Integrate with a vertical strip (dx)
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:
Figure 5.1.3(c) Integrate with a horizontal strip (dy)
The animation in Figure 5.1.3(c) shows a horizontal strip of thickness dy whose vertical sweep from y=0 to y=1 covers the complete region between the graphs of x=f−1y=y2 and x=g−1y=y. This corresponds to the integration in Table 5.1.3(c).
∫01y−y2 ⅆy = 13
Table 5.1.3(c) Calculation of the area between x=y2 and x=y
For integration with a horizontal strip between the curves x=Gy and x=Fy, the inequality Fy≥Gy must hold. If it does, then think of the functions as x=xLy and x=xRy, where xL defines the leftmost boundary; and xR, the rightmost. Accordingly, the area by horizontal strips is
∫yaybxRy−xLy ⅆy
where ya is the smallest y-value for the strip and yb is the largest.
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