Chapter 5: Applications of Integration
Section 5.1: Area of a Plane Region
Example 5.1.2
Calculate the plane area bounded by the graph of fx=x3−7 x2+5 x+4 and the x-axis.
Solution
Recall Example 4.2.5. where the area of a slightly different plane region was found. The shaded region in Figure 5.1.2(a) is bounded by y=fx and y=0. However, part of the region lies below the x-axis, the integration must be over two contiguous intervals, a,b and b,c, where a,b,c are the x-intercepts of f.
module () local f,q,p1,p2,p3,p; f:=x^3-7*x^2+5*x+4; q:=fsolve(f,x,complex); p1:=plot(f,x=q[1]..q[3],labels=[x,y],color=black,filled=[color=red,transparency=.7],thickness=2); p2:=plots:-textplot({[6,.7,typeset(c)],[-.3,.7,typeset(a)],[1.5,.7,typeset(b)]},font=[default,12]); p3:=plot([[q[1],0],[q[3],0]],style=line,color=black,thickness=2); p:=plots:-display(p1,p2,p3); print(p); end module:
Figure 5.1.2(a) Region bounded by fx and thex-axis
Define the function f
Control-drag fx=… Context Panel: Assign Function
fx=x3−7 x2+5 x+4→assign as functionf
Obtain the x-intercepts a,b,c
Write the equation fx=0 and press the Enter key.
Context Panel: Solve≻Numerically Solve
Context Panel: Conversions: To List
Context Panel: Assign to a Name≻x
fx=0
x3−7x2+5x+4=0
→solve
−0.4699899531,1.402749868,6.067240085
→to list
→assign to a name
X
Calculate the area of the region bounded by y=fx and y=0
Expression palette: Definite-integral template
Context Panel: Evaluate and Display Inline
∫X1X2fx ⅆx−∫X2X3fx ⅆx = 77.25663235
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