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The shading in Figure 4.3.1(a) corresponds to the area bounded by the -axis and the graph of on the interval . Because part of the shaded region is below the -axis, the area is given by
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Alternatively, evaluate the definite integral:
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Figure 4.3.1(a) Graph of on
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The antiderivative of 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 , the result would be the same. The value of at the upper limit is , as is the value at the lower limit; adding an arbitrary constant to the antiderivative induces in the evaluation of the antiderivative at the endpoints.
Finally, the two values and are not the same. If the graph of crosses the -axis in the interval of integration, its definite integral will be less than the area enclosed by this graph and the -axis.