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/material/simple/id/1091779
      Choose an upper AND lower function to define a region, then choose whether you want to use square, semi-circular, or triangular cross-sections. This GeoGebra app allows you to visualize what the region formed by the cross-sections looks like. Move the "xval" slider, and the solid object will fill in. Move the slider backward, and the "wire-frame" of the cross-sections remains. You can even increase the frequency of the cross-sections, if you like. The cool part is that it's really easy to move around and visualize the shape, too. Also notice that this app displays all of the integral set-up involved in finding the volume, so you can use this to check your work!
  • 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/m/1200891
      You can enter an upper/lower function, upper/lower bounds, and an axis of rotation, then move the slider (or hit the play button) to rotate the object. If you want to visualize a given solid formed by revolution, this app is your best bet.
    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).
    4. http://tube.geogebra.org/m/3787
      Pros:
      -Can illustrate Riemann sum aspect and make n --> infinity
      -Can enter in your function of choice (lower function is automatically x-axis)
      -It will actually rotate just the Riemann sum without showing the solid at the same time. (deselect "show revolving trace") That's kind of neat.
      -Animation tends to be very fluid for smaller n

      Cons:
      -Can't enter a lower function
      -Animation gets slower for large n
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.