Solve the Problem $x = a \cos t , y = b \sin t , 0 \leq t \leq 2 \pi$

Multiple Choice

Solve the problem.
-The length of the ellipse
$x = a \cos t , y = b \sin t , 0 \leq t \leq 2 \pi$ is $\text { Length } = 4 \mathrm { a } \int _ { 0 } ^ { \pi / 2 } \sqrt { 1 - \mathrm { e } ^ { 2 } \cos ^ { 2 } \mathrm { t } } \mathrm { dt }$ where $\mathrm { e }$ is the ellipse's eccentricity. Use Simpson's Rule with $n = 6$ to estimate the length of the ellipse when $a = 2$ and $e = \frac { 1 } { 3 }$ .

A) 12.1751
B) 12.2097
C) 11.8956
D) 11.2546