# “Reverse thinking” exam questions-reference request

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)?

• I'm not certain this question fits here, but I thought I would try. Also, could someone please Community Wikify this for me? – Brian Mar 19 '11 at 2:52
• Have you looked at the (nice!) book "Street-Fighting Mathematics"? I think some version is even available on-line. – Srivatsan Sep 1 '11 at 15:30