Chapter 4: Integration
Section 4.2: The Definite Integral
Example 4.2.5
Calculate the area bounded by the x-axis, fx=x3−7 x2+5 x+4, and the lines x=−1,x=2.
Solution
In Figure 4.2.5(a), the areas of regions A,B, and C, shaded in yellow, must be calculated separately because regions A and C are below the x-axis.
Thus, it is imperative to find the intercepts a and b.
Unfortunately, exact representations of these intercepts are large and cumbersome expressions. These intercepts will be approximated numerically.
Once the values of a and b have been determined, the desired area would be given by
∫abfx ⅆx−∫−1afx ⅆx−∫b2fx ⅆx
Figure 4.2.5(a) Graph of fx=x3−7 x2+5 x+4
Table 4.2.5(a) lists the exact values of a and b, obtained by simplifying the results returned by Maple's solve command.
λ=arctan9263 1087/3
a=7−102 sinλ−34 cosλ/3
b=7+102 sinλ−34 cosλ/3
Table 4.2.5(a) Exact values of the intercepts a and b
Numeric approximations of a and b are found below, and these values are used in the appropriate definite integrals to calculate the sum of the areas in regions A,B, and C in Figure 4.2.5(a).
Define the function f
Control-drag fx=… Context Panel: Assign Function
fx=x3−7 x2+5 x+4→assign as functionf
Obtain numeric values for a and b
Write fx and press the Enter key.
Context Panel: Solve≻Numerically Solve
Context Panel: Conversions≻To List
Context Panel: Assign to a Name≻r
fx
x3−7x2+5x+4
→solve
−0.4699899531,1.402749868,6.067240085
→to list
→assign to a name
r
Calculate the area
Write the sum of definite integrals determining the area. Use r1 for a and r2 for b.
Context Panel: Evaluate and Display Inline
∫r1r2fx ⅆx−∫−1r1fx ⅆx−∫r22fx ⅆx = 10.01212695
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