Problems Plus 13: Volume of an Infinite Number of Beads

7 months ago

In this video I determine the volume of an infinite string of beads formed by rotating an exponential and trigonometric function about the x-axis. The presence of the sine trig function means that the resulting 3D solid shape has nodes with a value of 0, hence forms what appears to be beads on a string. I first use the formula for disks to determine the volume of one bead, and then use an online integral calculator to solve the resulting volume integral. To calculate the total volume of the infinite number of beads, I do so using 2 methods. The first is simply expanding out the series and realizing it has a telescoping sum, which obtains our answer. The second method involves rewriting the volume of the n-th bead in the form of a Geometric Series, and I show that it is convergent and hence we can plug in our geometric series sum formula to obtain the total volume. I also play around with the amazing GeoGebra graphing calculator and discuss how to graph in 3D using the formula of a circle and our given function as the radius. Epic stuff!

The timestamps of key parts of the video are listed below:

Problem 13: Rotating a curve to form a 3D string of beads: 0:00
Solution to Part (a): Volume of the n-th bead: 1:01
- Graphing in 3D with GeoGebra Graphing Calculator: 1:49
- Formula for disks: 6:27
- Using an online integral calculator: 11:19
Solution to Part (b): Total Volume of Beads: Method 1: Telescoping Sum: 17:36
- Method 2: Geometric Series: 23:48

This video was taken from my earlier video listed below: