4. (Selections from Spivak, Problems 19.30-31.) Find the length of the following curves.
(a) The graph of $f(x) = \log(\cos x)$, for $x \in [0, \frac{\pi}{3}]$;
(b) the graph of $g(x) = \arccos(e^x)$, for $-\log 2 \le x \le 0$;
(c) $x = a \cos^3 t$, $y = a \sin^3 t$, for $t \in [0, 2\pi]$;
(d) $r(\theta) = a(1 - \cos \theta)$ for $0 \le \theta \le 2\pi$ (in polar coordinates).
Please accompany your calculation by a rough sketch of the curve.
