Question

# Find an arc length parameterization of the curve that has the same orientation as the given curve and for which the reference point corresponds to t=0. Use an arc length s as a parameter. r$t$ = 3$e^t$ cos $t$i + 3$e^t$sin$t$j; 0<=t<=$3.14/2$

135

likes
676 views

## Answer to a math question Find an arc length parameterization of the curve that has the same orientation as the given curve and for which the reference point corresponds to t=0. Use an arc length s as a parameter. r$t$ = 3$e^t$ cos $t$i + 3$e^t$sin$t$j; 0<=t<=$3.14/2$

Cristian
4.7
The derivatives are x'$t$ = \frac{d}{dt}$3e^tcost$ = 3e^tcost - 3e^tsint = 3\sqrt{2}cos$t+\frac{\pi}{4}$ y'$t$ = \frac{d}{dt}$3e^tsint$ = 3e^tsint + 3e^tcost = 3\sqrt{2}sin$t+\frac{\pi}{4}$ So norm of the derivative is ||r'$t$|| = \sqrt{x'$t$^2 + y'$t$^2} = 3\sqrt{2}e^t The arc length parametrization is s$t$ = \int_0^t ||r'$\tau$|| d\tau x$t$ = \int_0^t 3\sqrt{2}e^{\tau}d\tau= $3\sqrt{2}e^{\tau}]_0^t = 3\sqrt{2}(e^t-1$ So, the arc length parametrization is s$t$ = 3\sqrt{2}$e^t-1$
Frequently asked questions $FAQs$
What is the limit as x approaches pi/4 of $tan(x$ - 1)/$sin(x$ - cos$x$)?