I've ended up teaching a multivariable calculus course this quarter. Our first topic is the double integral. I didn't actually make any notes for myself the first day, and found myself improvising while trying to think up a practical scenario in which we'd want to do a double integral. What I ended up saying was something along the lines of "Imagine you're on a magical adventure in an enchanted kingdom beyond the Moon, when you're abducted by the vicious King of the Moon Men, who threatens to throw you into his Lake of Fire and Torment unless you can correctly compute the volume of said lake from its depth at every point."

On Wednesday I made very careful notes.

Today I slipped again, and explained to my students that, while integrating over a simple rectangle might be enough to save us from the clutches of the Emperor Tomato Ketchup and his Swimming Pool of Fear, we are faced with the King of the Moon Men and his curvilinear Lake, a much trickier proposition altogether.