# 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 =…