Chapter 4: Integration
Section 4.3: Fundamental Theorem of Calculus and the Indefinite Integral
Example 4.3.1
Calculate the area bounded by the x-axis, the graph of y=x3 and the vertical lines x=−1, x=2; then use the FTC to evaluate the definite integral ∫−12x3 ⅆx. Are these two quantities the same?
Solution
The shading in Figure 4.3.1(a) corresponds to the area bounded by the x-axis and the graph of y=x3 on the interval −1,2. Because part of the shaded region is below the x-axis, the area is given by
−∫−10x3 ⅆx+∫02 x2 ⅆx = 3512
Alternatively, evaluate the definite integral:
∫−12x3 ⅆx= x44]−12 = 24−−14/4=15/4
Figure 4.3.1(a) Graph of x3 on −1,2
The antiderivative of x3 is found by the Power rule: add 1 to the power and divide by the new power.
The simplest antiderivative has been used. If the antiderivative were taken as x4/4+C, the result would be the same. The value of C at the upper limit is C, as is the value at the lower limit; adding an arbitrary constant to the antiderivative induces C−C=0 in the evaluation of the antiderivative at the endpoints.
x44+C−12=244+C−−144+C=164−14+C−C=154+0=154
Finally, the two values 35/12 and 15/4 are not the same. If the graph of f crosses the x-axis in the interval of integration, its definite integral will be less than the area enclosed by this graph and the x-axis.
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