Let R Be the Region Bounded Between the Two Ellipses $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 2 ^ { 2 } } = 1$

Multiple Choice

Let R be the region bounded between the two ellipses $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 2 ^ { 2 } } = 1$ and $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 2 ^ { 2 } } = 4$ Use this change of coordinates $x=3 r \cos t, y=2 r \sin t$ for $r \geq 0,0 \leq t \leq 2 \pi$ to evaluate the integral $\int _ { R } \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) d A$

A) 240 $\pi$
B) 3240 $\pi$
C) 162 $\pi$
D) 1620 $\pi$
E) 1620