Answer
The difference in length between the outer and inner loops:
${s_{outer}} - {s_{inner}} \simeq 196.084$
Work Step by Step
Using a computer algebra system we plot the curve and note that the inner loop corresponds to the interval $0 \le \theta \le \pi $; whereas the outer loop corresponds to the interval $\pi \le \theta \le 2\pi $.
By Theorem 1 of Section 12.2, the length of the inner loop is
${s_{inner}} = \mathop \smallint \limits_0^\pi {{\rm{e}}^\theta }{\sin ^2}\theta {\rm{d}}\theta $
Using a computer algebra system we obtain ${s_{inner}} \simeq 8.856$.
The length of the outer loop is
${s_{outer}} = \mathop \smallint \limits_\pi ^{2\pi } {{\rm{e}}^\theta }{\sin ^2}\theta {\rm{d}}\theta $
Using a computer algebra system we obtain ${s_{outer}} \simeq 204.94$.
The difference in length between the outer and inner loops is
${s_{outer}} - {s_{inner}} \simeq 196.084$
