# Subtract the Hole from the Whole.

Sometimes I think textbooks are too rigorous. Behind every Riemann sum is a definite integral. So, authors routinely show how to solve an application of integration problem by developing the method starting from the Riemann sum and proceeding to an integral that give the result that is summarized in a “formula.” There is nothing wrong with…

# Visualizing Solid Figures 3

Volume by “Washers” Today I will show you how to visualize not just the solid figures but the disks and washers used in computing the volume using Winplot. The next post will show how to draw shells. Winplot is a free program. Click here for Winplot and here for Winplot for Macs. For the example we’ll…

# Visualizing Solid Figures 2

You have probably caught on by now that Winplot is my favorite computer graphing program. In addition to being great at drawing a quick graph, it is able to produce and rotate 3D images of, among other things, solids of rotation, and solids with regular cross-sections. In this post I will discuss how to do…

# Visualizing Solid Figures 1

The shape of various solid of rotation and solids with regular cross-sections of which beginning calculus students are required to find the volume are often difficult to visualize. This post and the next two will discuss some of the ways you can help your students become familiar with these shapes. Teachers often use these as…

# Does Simplifying Make Things Simpler?

I taught a class today on volumes of solid of revolution; specifically the ones with holes through them by the so-called “washer method.”  For this method you often see the formula Where R is “outside radius” and r is the “inside radius” and both are functions of x. It seems to me that this “simplified”…

# Area and Volume Questions

AP Type Questions 4 Given equations that define a region in the plane students are asked to find its area and the volume of the solid formed when the region is revolved around a line or used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every…

# Volumes of Revolution

Applications of Integration 3 – Volumes of Rotation In out last post we discussed volumes of figures with regular cross sections. Many common figures can be analyzed as some region being rotated around a line, possibly one of its edges. For example, the region between the x-axis and the segment with equation  from x =…