Hello there,

I'm tutoring somebody at the moment and I'm having some trouble with a few double integral questions that have cropped up in their past papers. I'm not sure whether or not there are errors in the questions that make them impossible to tackle or whether I'm just over thinking things, so I'm looking for some confirmation. I'll give you the question, my final answer and explain my issue, and hopefully you guys can clear things up.

1) Sketch the domain of integration and evaluate the integral

For this I can evaluate the integral and get an answer of , but I can't find a way to sketch the region.

I've drawn a graph with the lines , but there is no region enclosed by these curves, so how can the region be sketched/specified? Also, if no such region exists, how is it possible to evaluate the integral?

2) Evaluate

Here I am able to evaluate the integral and arrive at the result:

.

However, I'm slightly concerned that I'm getting a function of as a result rather than a numerical result... is this correct?

Either I'm correct and I'm over-thinking it, or it is an erroneous question (any double integrals I've seen in the past have the variable limits on the inner integral and the numerical limits on the outer integral, hence giving a numerical result), OR there is another way to deal with double integrals with variable limits on the outer integral that I'm not yet aware of. Which is it?

3) The height ( ) of the ceiling of an art gallery is defined by the surface:

The gallery wall is circular (in plan) and is defined by the equation:

For , derive an integral expression for the area of the wall between the limits .

For this one I'm not sure where to get started at all. It might not even be a double integral, but some sort of line integral or something? Anybody have any hints?