Chapter 9
Areas and Riemann Sums
T
his chapter provides some of the groundwork and motivation for antiderivatives.
Finding the area underneath a curve has real-world applications; however, for
many curves, finding the area is difficult if not impossible to do using simple geometry.
Here, you approximate the area under a curve by using rectangles and then turn to
Riemann sums. The problems involving Riemann sums can be quite long and involved,
especially because shortcuts to finding the solution do exist; however, the approach used
in Riemann sums is the same approach you use when tackling definite integrals. It’s worth
understanding the idea behind Riemann sums so you can apply that approach to other
problems!
The Problems You’ll Work On
This chapter presents the following types of problems:
✓ Using left endpoints, right endpoints, and midpoints to estimate the area underneath
a curve
✓ Finding an expression for the definite integral using Riemann sums
✓ Expressing a given Riemann sum as a definite integral
✓ Evaluating definite integrals using Riemann sums
What to Watch Out For
Here are some things to keep in mind as you do the problems in this chapter:
✓ Estimating the area under a curve typically involves quite a bit of arithmetic but
shouldn’t be too difficult conceptually. The process should be straightforward after
you do a few problems.
✓ The problems on expressing a given Riemann sum as a definite integral don’t always
have unique solutions.
✓ To evaluate the problems involving Riemann sums, you need to know a few summation
formulas. You can find them in any standard calculus text if you don’t remember them —
or you can derive them!
Areas and Riemann Sums
T
his chapter provides some of the groundwork and motivation for antiderivatives.
Finding the area underneath a curve has real-world applications; however, for
many curves, finding the area is difficult if not impossible to do using simple geometry.
Here, you approximate the area under a curve by using rectangles and then turn to
Riemann sums. The problems involving Riemann sums can be quite long and involved,
especially because shortcuts to finding the solution do exist; however, the approach used
in Riemann sums is the same approach you use when tackling definite integrals. It’s worth
understanding the idea behind Riemann sums so you can apply that approach to other
problems!
The Problems You’ll Work On
This chapter presents the following types of problems:
✓ Using left endpoints, right endpoints, and midpoints to estimate the area underneath
a curve
✓ Finding an expression for the definite integral using Riemann sums
✓ Expressing a given Riemann sum as a definite integral
✓ Evaluating definite integrals using Riemann sums
What to Watch Out For
Here are some things to keep in mind as you do the problems in this chapter:
✓ Estimating the area under a curve typically involves quite a bit of arithmetic but
shouldn’t be too difficult conceptually. The process should be straightforward after
you do a few problems.
✓ The problems on expressing a given Riemann sum as a definite integral don’t always
have unique solutions.
✓ To evaluate the problems involving Riemann sums, you need to know a few summation
formulas. You can find them in any standard calculus text if you don’t remember them —
or you can derive them!