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  1. M

    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. M

    Is this integration probelm right so far?

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

    Probably simple

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

    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. M

    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. M

    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. M

    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. M

    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. M

    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. M

    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. M

    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. M

    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. M

    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. M

    Online calculus of variations resource

    I can't promise anything, but this is a fairly decent list of online texts:
  15. M

    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.