0
$\begingroup$

I am working on this problem. Even some of the notation has me confused (the vectors $\vec i$ and $\vec j$).

Let $\vec r(t):=ae^{-bt}\cos(t)\vec i +ae^{-bt}\sin(t)\vec j$ where $a$ and $b$ are positive constants. The trace of $\vec r (t)$ is called the logarithmic spiral.

(a) Show that as $t \to +\infty $, $\vec r (t)$ approaches the origin.

(b) Show that $\vec r (t)$ has finite arc length on $[0,\infty)$.

  • 0
    Get rid of the $\cos^2$'s and (+) $\sin^2$'s! You are close..2012-02-16

1 Answers 1

1

The length is given, as the OP wrote, by $\int_0^\infty\sqrt{a^2b^2e^{-2bt}(\cos^2t+\sin^2t)+a^2e^{-2bt}(\cos^2t+\sin^2t)}\,\,dt=$ $=\int_0^\infty ae^{-bt}\sqrt{b^2+1}\,\,dt=\left.-\frac{a\sqrt{b^2+1}}{b}\,e^{-bt}\right|_0^\infty=\frac{a}{b}\sqrt{b^2+1}$