Area Between Two Curves
This content follows naturally from the definition of an integral. The factor that determines a student’s success will be his/her ability to accurately draw the curves that are being evaluated. Therefore, it is a good idea to include some review of plotting.
With practice, students will begin to get the hang of finding regions whose area they are calculating.
The next step is to recognize that this area is given by the area beneath the higher function minus the area beneath the lower function. So in the example above students should be able to add to their picture the equation:
Understanding and becoming good at this is really just a matter of practice. It is best to start with diagrams that are provided, and then later to give word problems where the students are asked to make their own diagrams.
There is a great deal of content covered in this lesson and it can be differentiated in a number of ways. The goal with every problem is to find some volume in three-dimensional space, and it should be made clear that there are always a variety of ways to carry this out. Usually there is one way that is simpler than others; however, students should be encouraged to try whatever occurs to them and see if it works.
Now, the biggest hurdle to using integration for finding volumes is drawing an accurate diagram or being able to visualize the exact volume that is of interest. Some students are naturally very good at visualizing three-dimensional shapes and manipulating them mentally. These students can usually rotate the objects in their minds to tell whether to use the disc method or shell method and which function should go where in the integral. However, for the students without this natural ability it is important to not frustrate them. For this reason the content of drawing accurate figures should be stressed and students who possess a unique ability for this should be asked to explain their reasoning to the class and present the problems.
To teach students how to find volumes it is best to carry out a detailed problem and then have students repeat. Have students put away their books and pencils and go over a volume of rotation problem in detail. Make sure everyone understands each step, and then have them do the problem on their own after erasing the board.
If there are a decent number of students that are very good at visualizing the volumes, it is a great idea to form groups where each group has at least one of these students. The groups can work a number of problems and the teacher should focus on making sure that everyone is participating and learning. Students who are ahead of the curve should be told that they are not done with a problem until everyone in the group understands it as well as they do.
Naturally, students can be tested on this material by having them calculate the volume of a number of different shapes. It is a good idea to start with problems where the shape is provided and then move into problems which are more abstract.
and rotating it around the x-axis:
Drawing careful diagrams of the discs that are being used will be very helpful:
The Length of a Plane Curve
The formula for arclength:
will seem awkward to many students at first, but it can be explained on geometric grounds. But before doing this, it is a good idea to connect the concepts of integral and summation once more.
Now back to the arc-length of a curve. Intuitively, the total length of any curve is equal to the sum of tiny “pieces” along the curve. So if we zoom way in on one of these pieces it might look like the following:
So summing up these tiny segments means taking the integral of this and we are left with the given formula for arc-length.
Then students can practice integrating:
Area of Surface of Revolution
The formulas used here relate closely to the formula for the circumference of a circle. In teaching this material the intuitive nature of the concepts should not be disregarded. The area of a surface of revolution is given essentially by summing up the circumferences of all the circles that make it up. Think about a high stack of thin tires. The surface that they form can be thought of as a combination of all of their circumferences, and so we expect to just integrate over these values in order to find its area.
As with the volume calculations, the biggest hurdle to overcome in mastering this material is being able to draw a very accurate picture. Students should be begin with practice problems where they are told this is now an art class. They should then be given three positive functions like:
Once they have completed this, students can be paired and each partner can be given a curve to be rotated and they should guess which have the larger area. Then each should calculate the area for his/her curve and they should compare to see if their guess was correct.
Applications from Physics, Engineering and Statistics
When the pressure is a constant along one of the two directions (like in the example) then this becomes a single integral.
Probability densities will also seem very natural if introduced properly. For example, it’s a good idea to consider the following examples:
a. What is the probability of obtaining any given number in particular?
b. If we were to plot the probability as a function of the possible outcome, what would it look like?
c. Now suppose we have a dartboard and somebody who throws a dart randomly. What would the plot look like for the probabilities here: