Chapter 10
The Fundamental Theorem of Calculus
and the Net Change Theorem
U
sing 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.
The Fundamental Theorem of Calculus
and the Net Change Theorem
U
sing 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.