Length, Area, and Volumes
Calculus is really the study of two items—measuring how fast something is changing and accumulating material a little bit at a time. You’ve had a fair amount of experience with the measuring rates of change. In Part 2, we take some time to work on the accumulation. Keep in mind, we are still looking at mathematics through the microscope. That means what might not look like much of a region to the naked eye can be rather formidable when we zoom in on it.
You’re asked to use your imagination when we look at volumes. First, we begin working in a two-dimensional plane but then we rotate figures around vertical and horizontal lines. You then are asked to stand “at the end of the line” and look at a cross section. (Think of cutting a piece of fruit with a sharp knife and turning what was an interior part of the fruit toward you.) Getting the vision of the cross section in your mind goes a long way to helping you work with volumes.
The Pythagorean Theorem also comes into play again as you use it in very small increments to measure the length along a curve. You also learn about the area formulas for rectangles and trapezoids used to help find areas of regions with curved boundaries.