Volume Visualization Resources:
If you are ever having trouble visualizing what a solid object looks like, these apps will help you to see what's going on. Read the descriptions below to get an idea of which app fits your needs best. Otherwise, happy visualizing!
  • Solids by Cross-Sections
    1. http://4.bp.blogspot.com/-O98O8VV_PgE/UdNO32JJCsI/AAAAAAAAA0c/Pi8ZNa8g73s/s1600/cross+sections.jpeg
      Here are some examples of solids formed with cross-sections lined up on top of a 2D region. Notice that the shapes of your solids can vary wildly if you change your cross-section shape.
    2. An AWESOME GeoGebra app for showing cross-sections: https://www.geogebra.org/classic/XArpgR3A
  • Solids by Revolution - Note that none of these are perfect. They all do certain things well but, there isn't one "best" app to illustrate solids by revolution.
    1. https://www.geogebra.org/classic/gFXdBH5J
    2. http://www.geogebra.org/student/m1066127
      This app doesn't let you enter in your own function but instead has 4 fixed functions you can switch between. My favorite part about this one is that this illustrate the Riemann sum involved in finding the volume of the 4 solids - you can even vary n (the number of cross-sections/sub-intervals) to see how the volume approximations become exact as n tends to infinity. Another plus: There's both a 2-D view of the area under the curve, and then a 3-D view of what happens when the area is rotated about the axis.
    3. https://www.geogebra.org/m/25298
      This app has 3 specific functions you can toggle between. You can vary the interval of interest and see how that changes the solid generated. Because this app only focuses on 3 functions, the animation for those 3 functions is very clean, and there aren't many stray lines left over after each rotation is animated (as is the case with app 1).
Disclaimer about all of the above apps:
One catch on a lot of these apps is that a lot of the lines and/or cross-sections remain visible even when you switch the various parameters. `For that reason, you kind of need to refresh after every single visualization. Still, it's a neat tool to help students start to see these 3-D shapes! The "wire frame" of the cross-sections still being visible can also be helpful to illustrate what we're doing sometimes.