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The Fundamental Theorem of Calculus, Part 2 (also known as the evaluation theorem ) states that if we can find an antiderivative for the integrand, then we can evaluate the definite integral by evaluating the antiderivative at the endpoints of the interval and subtracting.
Let be a regular partition of Then, we can write
Now, we know F is an antiderivative of f over so by the Mean Value Theorem (see The Mean Value Theorem ) for we can find in such that
Then, substituting into the previous equation, we have
Taking the limit of both sides as we obtain
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Recall the power rule for Antiderivatives :
Use this rule to find the antiderivative of the function and then apply the theorem. We have
Analysis
Notice that we did not include the “+ C ” term when we wrote the antiderivative. The reason is that, according to the Fundamental Theorem of Calculus, Part 2, any antiderivative works. So, for convenience, we chose the antiderivative with If we had chosen another antiderivative, the constant term would have canceled out. This always happens when evaluating a definite integral.
The region of the area we just calculated is depicted in [link] . Note that the region between the curve and the x -axis is all below the x -axis. Area is always positive, but a definite integral can still produce a negative number (a net signed area). For example, if this were a profit function, a negative number indicates the company is operating at a loss over the given interval.
Evaluate the following integral using the Fundamental Theorem of Calculus, Part 2:
First, eliminate the radical by rewriting the integral using rational exponents. Then, separate the numerator terms by writing each one over the denominator:
Use the properties of exponents to simplify:
Now, integrate using the power rule:
See [link] .
James and Kathy are racing on roller skates. They race along a long, straight track, and whoever has gone the farthest after 5 sec wins a prize. If James can skate at a velocity of ft/sec and Kathy can skate at a velocity of ft/sec, who is going to win the race?
We need to integrate both functions over the interval and see which value is bigger. For James, we want to calculate
Using the power rule, we have
Thus, James has skated 50 ft after 5 sec. Turning now to Kathy, we want to calculate
We know is an antiderivative of so it is reasonable to expect that an antiderivative of would involve However, when we differentiate we get as a result of the chain rule, so we have to account for this additional coefficient when we integrate. We obtain
Kathy has skated approximately 50.6 ft after 5 sec. Kathy wins, but not by much!
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