This week we did an activity about finding the volume of a solid created by some equations using known cross-sections. We did an arts and crafts activity to help us learn about this and visualize the problems. I found this activity pretty helpful, I guess.
Basically, to find the volume of the solid, you have to know the equations and the shape of the cross sections. These cross-sections can be any shape, and the volume depends on what shape they are. To find the volume, you just have to take the integral of the area of the cross-section, generalized using the equations. This means that you can use any shape of cross section as long as you know how to find the area.
To find the area, you need to plug in the distance between equations and such into the area formula for the shape of the cross section. Usually the base of the cross sections, or the diameter if they are a circular, will be one equation minus the other.
After having done this activity, it is pretty easy to visualize what a solid created with known cross-section would look like; it was helpful in that way. Even if you can’t visualize the solid, that is okay as long as you can find the area of the cross sections.
I think this activity was helpful in helping me to understand what was being asked and what the solids were/what they looked like. Now I feel that I can confidently do most known cross sections problems.
AP Calc student. I write these blogs.