Area Between Two Curves
One curve is completely above the axis, the other is below. Ask the students, What do you think will happen? To calculate:
It is valuable have a conceptual understanding of the idea that cross sectional areas added together allows for the calculation of volumes. There are more methods and formulas than one can reasonably remember, although some common, or maybe difficult ones, are worth the time. There are many questions outside of these forms, however, that are favorites on many tests. One that frequently gets chosen is asking for the volume of the solid that has a specified base, with a particular shape above that base. Here is an example:
A picture is very helpful in organizing all the information. The first order of business is to figure out what the area is that is needed to iterate to get the volume requested. The half circles that are shaded darker are the area in question, so they are what we need to figure out the expression for the area of those shapes next.
It's worth making it into a mantra: “Find volumes by integrating areas for the length of the solid.”
The Length of a Plane Curve
When using a computer solver, the key is to make sure that the derivatives are taken correctly, and that the input syntax is correct.
Find the length of the line described by the parametric equations
Area of a Surface of Revolution
Newton’s Law of Cooling states that the rate of temperature change is equal to the heat transfer coefficient times the surface area times the difference in temperatures. Stated in variables:
Here, we need to substitute all the information we have into the Law of Cooling function. This is a little different than normal, as we are not asking to compute the area of the surface, but need to state where the limits of integration are to get the proper area needed to conform to the requirements. Because the integral is going to take up some serious space, we should first solve for the total minimum area.
Now setting the integral equal to this quantity, but leaving the variable we need to solve for in the upper limit:
Applications from Physics, Engineering and Statistics
These problems are really illustrative of how calculus was developed and the questions that drove the techniques and theorae learned thus far. Problems that have natural or applied motivation often do not work as “cleanly” as the types of packaged problems typically presented in textbooks for practice. There are a few tools that are helpful in navigating these problems.
- Always keep track of vectored quantities. While it is sometimes a bit of extra work to make sure all the signs are set up in the correct manner, one nice result of doing careful work up front is that the answer falls with the correct sign with only doing the correct math.
- When it doubt, write all the units, all the time. Sometimes the units illustrate the next step and can keep you going when stuck. For example: finding quantities like work involves multiplying two other quantities. If you have force as a function of distance, then the product of the two is area, indicating that an integral is called for.
- Use significant space for work. Some problems or formulae may use odd numbers or expressions that can be confusing if they are crammed into a small space. I am thinking specifically about the standard normal distribution, which has a complex exponent that is easy to get mixed up.
- Draw a picture. Always. The quality of a picture, as well as the labeling of quantities is imperative for keeping track of necessary information, and how the quantities relate.