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1. Volume vs. Area of a Surface of Revolution

Off the top of my head (that is, take this with a huge grain of salt), I think the approximating surface doesn't really matter in the answer, but the cone/cylinder might give the simplest (or maybe easiest to visualize) way to get to the answer.
2. Is this integration probelm right so far?

Infinity is just \infty. Dunno about the other. Anyway, it looks correct to me.
3. Probably simple

Just solve for the constant here: -1/2y^2=t+C.
4. Limit problem

What do you mean you "moved it down"? Regardless, I suggest rewriting e^{-x^{2}} using the facts you know about negative exponents. Probably easier that way.
5. Complex Integral

I believe this does require integration along a contour. I think it goes something like this: \int_{-\infty}^{\infty}\frac{\ln{(a+ix)}}{x^2+1}dx = \int_{\gamma + \sigma} \frac{\ln{(a+ix)}}{x^2+1}dx + \int_{-\sigma}\frac{\ln{(a+ix)}}{x^2+1}dx where \gamma is the contour from -R to R along...
6. Trouble understanding

I think the point of saying that such a neighborhood shouldn't exist is that it's possible for f(u) to be something like 1/(u - Uo). But if u(x) is identically Uo in a neighborhood of a, the limit can't exist. But I think if we assume that f(u) has at worst a removable singularity at Uo, then...
7. Integration Question

Try to write the integrand as sin(x) * f(cos(x)) or cos(x) * f(sin(x)), where f is some algebraic function.
8. The partial derivatives of arctan(y/x)

That looks right. To get the other, just take d/du(arctan(u)) * du/dx, where u = y/x, just like in the previous situation.
9. Divergence Theorem - Confused :s (2 problems)

Technically, what you should do is find a tangent vector for your circle, then get a vector perpendicular to that to find the normal. But, as far as I remember, taking the gradient is essentially a cheat. Basically, what you're doing is saying that every level set of f(x,y) = x^2 + y^2...
10. Quick integral question

Most tables of integrals have a reduction formula for those kinds of integrals. But that's not exactly time saving either.
11. Divergence Theorem - Confused :s (2 problems)

Well, the big problem with the first part you're having trouble with is that f(x) = (1 - x^2)^(1/2) only represents the top part of the circle. The second problem is that d(sigma) isn't d(theta).
12. L'Hopital's Rule - I'm loosing my hair

As x -> 0, -x -> 0. As x -> 0, sqrt(4 - x^2) -> 2. Also, be careful about talking about equivalence here. L'Hospital's Rule just says that, if you have f(x)/g(x) such that f(x) -> 0 and g(x) -> 0 as x -> a, then f(x)/g(x) approaches the same limit as f'(x)/g'(x) as x -> a, if said limit exists.
13. L'Hopital's Rule - I'm loosing my hair

The only thing I see wrong with what you got is that the derivative of 4 - x^2 is -2x. But then again, I'm half asleep.
14. Online calculus of variations resource

I can't promise anything, but this is a fairly decent list of online texts: http://www.geocities.com/alex_stef/mylist.html
15. A question professor couldnt solve!

Yeah, the intermediate value property guarantees no discontinuities of this type. You can have discontinuities where the limit fails to exist though.