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Using Limits and Riemann Sums to Evaluate Definite Integrals
541–545 Use the limit form of the definition of the integral to evaluate the integral.
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Chapter 10
The Fundamental Theorem of Calculus and the Net Change Theorem
Using Riemann sums to evaluate definite integrals (see Chapter 9) can be a cumbersome process. Fortunately, the fundamental theorem of calculus gives you a much easier way to evaluate definite integrals. In addition to evaluating definite integrals in this chapter, you start finding antiderivatives, or indefinite integrals. The net change theorem problems at the end of this chapter offer some insight into the use of definite integrals.
Although the antiderivative problems you encounter in this chapter aren’t too complex, finding antiderivatives is in general a much more difficult process than finding derivatives, so consider yourself warned! You encounter many challenging antiderivative problems in later chapters.
The Problems You’ll Work On
In this chapter, you see a variety of antiderivative problems:
Finding derivatives of integrals
Evaluating definite integrals
Computing indefinite integrals
Using the net change theorem to interpret definite integrals and to find the distance and displacement of a particle
What to Watch Out For
Although many of the problems in the chapter are easier antiderivative problems, you still need to be careful. Here are some tips:
Simplify before computing the antiderivative. Don’t forget to use trigonometric identities when simplifying the integrand.
You don’t often see problems that ask you to find derivatives of integrals, but make sure you practice them. They usually aren’t that difficult, so they make for easier points on a quiz or test.
Note the difference between distance and displacement; distance is always greater than or equal to zero, whereas displacement may be positive, negative, or zero! Finding the distance traveled typically involves more work than simply finding the displacement.
Using the Fundamental Theorem of Calculus to Find Derivatives
546–557 Find the derivative of the given function.
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Working with Basic Examples of Definite Integrals
558–570 Evaluate the definite integral using basic antiderivative rules.
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