Multiple Choice 1 – 9 were generally ok. Only a few correct responses to 10. Read it over carefully and consider the whole point of transforming an x, y integral into an s, t integral.

Problem Solving

- For the most part, this was straightforward. It was surprising how many students didn’t find the x= 0 and y = 0 solutions to setting the first partial derivatives equal to 0. I think this means more practice with algebra.
- Lots of trouble with seeing why the volume of the box was the quantity given. Draw a box, imagining that one corner is at the origin and the other corner on the given plane. What will the volume of that box be, in terms of x, y, z? Solve the equation of the plane for z, and then substitute into the V expression. A surprising number of students who attempted (and even were successful in question 1) didn’t even attempt part b), even though the function was given and the context was finding a local maximum.
- Nearly everyone evaluated the iterated integral correctly – yay! Fewer were able to draw the region, and fewer still found the area of the region to check their answer to part a).
- Lots of success. Choosing a good order and using u-substitution were key.
- This problem revealed significant gaps in most students. It will help (and perhaps even be essential) to draw the region in the x-y plane and use this to help determine the bounds (limits) of integration. Good practice with integration and algebra.
- Most realized that polar coordinates were the way to go, and were able to see that the bounds on r were square root (2) to 2 sin theta. There was more difficulty with finding the bounds for theta – look at the sketch to find the bounds; also remember that theta is measured counterclockwise from the positive x-axis. Students wound up needing to find an integral of sin^2; several had not written down the relevant trig identity and didn’t feel like they could ask. However, most of you know Euler’s formula from physics – there’s a pretty derivation of some useful trig identities; worth learning it so this doesn’t happen again. A somewhat surprising mistake was to have the wrong order of integration, such that there was a variable in the outer limit.
- This problem also revealed significant gaps. Note that it can be addressed either in cylindrical coordinates or spherical coordinates; each has their advantages. More good practice with integration and algebra.
- Surprisingly difficult, even with the hint. Try a coordinate change, with s = x/a and t = y/b.
- I was pleased with many responses. I tried to be as generous as I could, but needed there to be some math content, and some calculus-specific content.

Bonus: no-one made much useful progress on the bonus. The napkin ring problem is a famous one. The result is amazing: the volume of the napkin ring depends only on the height h! What?!?!