Chapter 11
Applications of Integration
T
his chapter presents questions related to applications of integrals: finding the area between
curves, finding the volumes of a solid, and calculating the work done by a varying force. The
work problems contain a variety of questions, all of which apply to a number of real-life situations and relate to questions that you may encounter in a physics class. At the end of the chapter, you answer questions related to finding the average value of a function on an interval.
The Problems You’ll Work On
In this chapter, you see a variety of applications of the definite integral:
✓ Finding areas between curves
✓ Using the disk/washer method to find volumes of revolution
✓ Using the shell method to find volumes of revolution
✓ Finding volumes of solids using cross-sectional slices
✓ Finding the amount of work done when applying a force to an object
✓ Finding the average value of a continuous function on an interval
What to Watch Out For
Here are a few things to consider for the problems in this chapter:
✓ Make graphs for the area and volume problems to help you visualize as much as possible.
✓ Don’t get the formulas and procedures for the disk/washer method mixed up with the
shell method; it’s easy to do! For example, when rotating regions about a horizontal
line using disks/washers, your curve should be of the form y = f(x), but if you’re using
shells, your curve should be of the form x = g(y). When rotating a region about a
vertical line and using disks/washers, your curve should be of the form x = g(y), but if
you’re using shells, your curve should be of the form y = f(x).
✓ Some of the volume of revolution problems can be solved using either the disk/washer
method or the shell method; other problems can be solved easily only by using one
method. Pay attention to which problems seem to be doable using either method and
which ones do not.
✓ The work problems often give people a bit of a challenge, so don’t worry if your first
attempt isn’t correct. Keep trying!
Applications of Integration
T
his chapter presents questions related to applications of integrals: finding the area between
curves, finding the volumes of a solid, and calculating the work done by a varying force. The
work problems contain a variety of questions, all of which apply to a number of real-life situations and relate to questions that you may encounter in a physics class. At the end of the chapter, you answer questions related to finding the average value of a function on an interval.
The Problems You’ll Work On
In this chapter, you see a variety of applications of the definite integral:
✓ Finding areas between curves
✓ Using the disk/washer method to find volumes of revolution
✓ Using the shell method to find volumes of revolution
✓ Finding volumes of solids using cross-sectional slices
✓ Finding the amount of work done when applying a force to an object
✓ Finding the average value of a continuous function on an interval
What to Watch Out For
Here are a few things to consider for the problems in this chapter:
✓ Make graphs for the area and volume problems to help you visualize as much as possible.
✓ Don’t get the formulas and procedures for the disk/washer method mixed up with the
shell method; it’s easy to do! For example, when rotating regions about a horizontal
line using disks/washers, your curve should be of the form y = f(x), but if you’re using
shells, your curve should be of the form x = g(y). When rotating a region about a
vertical line and using disks/washers, your curve should be of the form x = g(y), but if
you’re using shells, your curve should be of the form y = f(x).
✓ Some of the volume of revolution problems can be solved using either the disk/washer
method or the shell method; other problems can be solved easily only by using one
method. Pay attention to which problems seem to be doable using either method and
which ones do not.
✓ The work problems often give people a bit of a challenge, so don’t worry if your first
attempt isn’t correct. Keep trying!