Volumes of Revolution Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Finding the volumes of shapes produced by revolving a curve around one of the axes is one popular use for the slice method for finding volumes. The “Solids of Revolution” and “The Disk Method” subsections in section 6.2 of our textbook explain this idea and give examples. The following problems give you a chance to practice and discuss the idea.

Imagine taking the graph of y = 1/x between x = 1 and x = 2, and rotating it around the x axis to create a shape that looks vaguely like a trumpet. What is the volume of this trumpet?

What if you take the graph of y = x² + 1 between x = -1 and x = 1 and rotate it around the x axis? What is the resulting volume?

And here’s a little more challenging variation: what if you take the graph of y = x² + 1 but rotate it around the y axis? What’s the volume of that volume of revolution, between y = 1 and y = 2?