So, I'm preparing for an exam and I'm stuck with problems with volumes of solids of revolution. I have two examples:
a, find the volume of a solid, described as: $T=\{(x,y,z) : x^2 + y^2 - z^2 \leq 1, |z| \leq 1 \}$
b, find the volume, where $\{[x,y] \in \mathbb R^2; 0 \leq x \leq 1 \ \land 0 \leq y \leq \sqrt x \ arctg(x)\}$ is rotated around the axis $x$
As, I'm before an exam, I would really - really like a guide on how to solve problems like these. Or, if you help me with these problems (what you think it's the easiest method?), I believe I can work something out myself.
Any hint is appreciated, thanks in advance.
a. $V = \displaystyle \int_{-1}^1 \int_{-\sqrt{1+z^2}}^{\sqrt{1+z^2}} \int_{-\sqrt{1+z^2-y^2}}^{\sqrt{1+z^2-y^2}} 1 dxdydz$
Alternately, you can use polar coordinates with $r = \sqrt{1+z^2}$. Then set $x = r\cdot cos\theta$, and $y = r\cdot sin\theta$, then you have:
$V = \displaystyle \int_{-1}^1 \int_{0}^{2\pi} \int_{0}^{\sqrt{1+z^2}} r\cdot dr\cdot d\theta\cdot dz$
b. Use the disk method: your cross section area is: $\pi\cdot r^2 = \pi\cdot x\cdot (tan^{-1}x)^2 = f(x)$. Thus you simply integrate over the unit interval $[0,1]$:
$V = \displaystyle \int_{0}^1 f(x)dx$. To do this you can use integration by part.