One of the major applications of integration is to find the volumes of various solid figures.

Volume of Solids with Regular Cross-sections  This is where to start with volume problems. After all, solids of revolution are just a special case of solids with regular cross-sections.

Volumes of Revolution

Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

Visualizing Solid Figures

• Visualizing Solid Figures 1 Making models et al.
• Visualizing Solid Figures 2  Using Winplot to visualize solids with regular cross-sections.
• Visualizing Solid Figures 3 Using Winplot to visualize the Disk and Washer methods.
• Visualizing Solid Figures 4 Using Winplot to visualize finding volumes by cylindrical shells. While no problem on the AP Calculus exams will require a solution by the Cylindrical Shell method, students may use the method and be eligible for full credit. So this is “enrichment.” 
• Challenge – the half-full problem comparing the disk and shell methods and Visualizing Solid Figures 5  the detailed solution   

Why you Never Need Cylindrical Shells

Painting a Point

Revised and update October 22, 2018