Suppose $a>0$ is a constant. Let $C$ be the curve $y=\cosh x$, for $-a \leq x\leq a$. Let $D$ be the region bounded by $C$, $|x| = a$ and the $x$-axis.
1) Find the length of $C$
2) Find the area of the surface obtained by rotating $C$ about the $x$-axis
3) Find the volume of the solid obtained by rotating $D$ about the $y$-axis
For 1), is it $\sinh a$ ?
For 2), is it $2\pi a + \pi \sinh (2a)$ ?
Would anyone mind telling me how to solve the above problems? I really have no idea.
For (1) the arc length of C is given by
$L = \int_{-a}^a\sqrt{1+[y'(x)]^2}dx = 2\int_0^a\sqrt{1+\sinh^2(x)}dx =2\int_0^a\cosh(x)dx = 2\sinh(a) $