I am interested in exam questions that are "backwards" from how they are usually asked. For example:
Brian and Megan have the following question on their exam:
Find the volume of the solid obtained by rotating the region bounded between $y=x^2$ and $x=y^2$ about the $x$-axis.
Megan's integral looks like this: $2\pi \int_0^1 y\, (\sqrt{y}-y^2)\, dy$
Brian's integral looks like this: $\pi \int_0^1 {(\sqrt{x}-x^2)}^2\, dx$
When they evaluate their integrals they get different answers. Who is wrong? What is his or her mistake?
Or
Express $\displaystyle \lim_{n \to \infty} \frac{1}{n} \sum_{i=0}^n \frac{1}{1+(\frac{i}{n})}$ as a definate integral.
Does anyone have suggestions for where to look for more of them, research on their effectiveness, or even a good name for them (so I can search for them)?