Volumes of Revolution

We just started volumes in Calculus today – it is one of my favorite topics. Learning this material is one of my favorite memories from high school.

My son seemed to enjoy it, too. We worked through the 6 examples at the start of the section together this morning. When he got home from school I had him try a few examples on his own.

He picked two problems from the book. The first problem was finding the volume when the area between $y = x^2$ and $y^2 = x$ is rotated around the x-axis:

The second problem he chose asked him to find the volume of a “cap” of a sphere – this is both a really neat problem and a pretty challenging one. I was surprised that he chose this one, but it was fun to talk through:

For the final problem we worked in Mathematica and made a 3d picture of the volume created when you rotated the curve $y = \sin(x)$ between $x = 0$ and $x = \pi$ around the x-axis:

I’m excited to pick a few more of these shapes to print as we work through this topic.