One of the hardest topics for Calculus students to understand in the finding of volume when they are presented with the base of the region and a description of the cross section cut perpendicular to the base.  

I designed a lab activity to help students visualize these shapes and to also notice that all these problems are very similar to each other.  

Using the region bounded in the first quadrant by the x-axis, x=4, and the curve square root of x, students are given 8 different cross sectional shapes to actually build a model the solid they will find the volume of.  After students have glued down cross section to the region they then work together to find the volume of the solid using integration.  Through this activity of building the integrals, the students notice that all 8 shapes follow the same procedure.  Then they get to compare their volumes and notice there is a relationship between many of the volumes.  

Calculus: Volumes of Solids with Known Cross Sections - An Exploration

Here is what a few buyers said about the activity:

• Excellent activity. I laminated the shapes pages, students then cut out the pieces and used glue sticks to create the "volume". Students answered the questions pertaining to their created shape then traded with others so they saw all shapes. Students sailed through the practice problems the next day. This activity provided the visuals they needed to understand the concept.
• Great way to approach a challenging topic in AP Calculus.
• Love the visuals. I used to have my students do this with pipe cleaners, this is much more accurate.