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Chapter 4: Integration
Section 4.3: Fundamental Theorem of Calculus and the Indefinite Integral
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Essentials
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Fundamental Theorem of Calculus
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The basic mechanism for evaluating a definite integral is the Fundamental Theorem of Calculus (FTC), formally stated as Theorem 4.3.1.
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Theorem 4.3.1: Fundamental Theorem of Calculus
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For , is any antiderivative on
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is continuous on , differentiable on , and
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The first conclusion of the FTC is that a definite integral, the limit of a Riemann sum, can be evaluated by subtracting the endpoint values of any antiderivative. That is the meaning of the symbols in the first conclusion of the theorem, where is any antiderivative of the integrand , and the value of the definite integral is .
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The second conclusion of the FTC is that a definite integral with a varying endpoint defines a function whose derivative is the integrand. In other words, the integral in hypothesis (3) defines an antiderivative, so that integration (the limit of a Riemann sum) and differentiation are inverse operations.
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Because of the FTC, and only because of the FTC, the integral sign is used to denote the operation of finding an antiderivative. Thus, the antiderivative is identified with the indefinite integral, denoted by .
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For the functions in Table 4.3.1, Maple computes their definite integrals on by taking the limit of a right Riemann sum. The results are consistent with finding an antiderivative and evaluating , that is, by using the FTC.
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Limit of right Riemann sum
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Table 4.3.1 Evaluation of via Riemann sum is consistent with FTC
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The first four functions all yield to the "Power rule for antidifferentiation,"
that is, to the rule "add 1 to the power and divide by the new power." (In symbols, this is the third entry in Table 3.10.1.) Antiderivatives for the exponential and trig functions can also be found in Table 3.10.1.
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If is an antiderivative of , a common notation for the difference that arises in the evaluation of the definite integral is . (Sometimes, the stroke is replaced by "", the closing square bracket.) Although such notation can sometimes be written in Maple, it is not part of the code structure used when a definite integral is evaluated by Maple.
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From Riemann Sum to Antiderivative: A Sketch
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The following is a sketch of a demonstration as to why the limit of a Riemann sum should result in the difference in the endpoint values of an antiderivative of the integrand. The essential link is the Mean Value theorem stated in the form for some between and .
Consider a right Riemann sum for the derivative on the interval . Apply the Mean Value theorem to each subinterval , selecting as the point to evaluate as the for which the equality
holds, . In the Riemann sum, replace each term with so that the Riemann sum becomes
or
which collapses to .
In the limit as and , the need to pick the to satisfy the Mean Value theorem disappears since each subinterval shrinks to zero length, and there must always be an appropriate in every such shrinking subinterval.
In the definite integral (which is just the limit of the Riemann sum) where the integrand is the derivative , the value of the definite integral is the difference , the difference in the endpoint values of the antiderivative.
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Examples
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Example 4.3.1
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Calculate the area bounded by the -axis, the graph of and the vertical lines , ; then use the FTC to evaluate the definite integral . Are these two quantities the same?
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Example 4.3.2
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Use the FTC to evaluate the definite integral .
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Example 4.3.3
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Obtain an explicit rule for the function ; then show is the integrand evaluated at .
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Example 4.3.4
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Use Maple to obtain an explicit rule for the function ; then show is the integrand evaluated at .
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Example 4.3.5
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Example 4.3.6
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Evaluate the indefinite integral .
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