Fun with Calculus: Modeling Volume of Solids with Known Cross Sections (Part 1)

Using the model of the area in between two curves from the previous post, I decided to create solid with known cross sections.  Information about the volume of solids with known cross sections can be found here or there.

For my project, I chose a square cross section.  The idea behind cross sections is that each slice has an infinitely thin thickness.  However, yarn is not infinitely thin, and crocheting a square for every single slice that can be made takes up a lot of yarn, so I only made 15 slices to be evenly spaced out.  Just keep in mind that there will be spaces in between each cross section in my model, spaces which should not exist.

Each square cross section I made had a side length that corresponded with the difference between the y-values for f(x) and g(x) at different x-values.

First cross section I made

Another, bigger cross section

 I will attach each cross section on perpendicular to the base, which is the model of the area between the curves f(x) and g(x).

An idea of how each cross section will be attached

Each cross section I made generally increased in size, since as the x-value increased, the y-value generally increased until the vertex.  After, each cross section decreased in size.

5 of the cross sections I made

To avoid confusion before I attached each square to the base, I stacked the squares up in increasing side length.  The two piles represent the squares that belong to the portion of the solid that is to the left of the vertex and the remaining squares that belong on the other portion of the solid that is to the right of the vertex.

All of the square cross sections I made

To see the final project, go to Part 2.