Couple Calculus Problems (Mostly Solid Revolutions)

1. Sep 30, 2008

myanmar

1. Find the vol. of a solid generated by revolving the region bounded by $$y = 0$$ and $$= \frac {1}{\sqrt {x}}$$ for $$1 \le x \le 2$$ about the line $$y = - 1$$
2. Find the volume of the solid generated by revolving the region bounded by the lines $$x=0$$, $$y = 0$$, $$x = 1$$, and the curve $$y = e^{x^{2}}$$. about the $$y$$ - axis
3. Can you write the func. $$f(x) = x$$ as the product of two differentiable functions ($$g(x)$$ and $$h(x)$$) if $$g(0) = h(0) = 0$$.
4. Show that if
$$\frac {a_{0}}{1} + \frac {a_{1}}{2} + ... + \frac {a_{n}}{n + 1}$$
then
$$a_{0} + a_{1}x + ... + a_{n}x^{n} = 0$$
for some x in $$[0,1]$$
5. Find the area bounded by $$y = \sqrt {x}$$, $$\frac {1}{x}$$, $$\frac {1}{x^{2}}$$
6. The region in the first quadrant bounded by $$x = 2$$ and $$y = 4$$ and $$x^{2} = 4y$$ is revolved around the $$y$$-axis. Find the volume.

Attempts at solution (so far)
3. Yes. But I don't know how to show this. It's true for $x^{\frac {1}{2}}$ and $x^{\frac {1}{2}}$ or $x^{\frac {1}{3}}$ and $x^{\frac {2}{3}}$.