The Iterated Integral $\int_{0}^{2 \pi} \int_{0}^{2} \int_{0}^{\sqrt{4-r^{2}}} r e^{z} d z d r d \theta$

Multiple Choice

The iterated integral $\int_{0}^{2 \pi} \int_{0}^{2} \int_{0}^{\sqrt{4-r^{2}}} r e^{z} d z d r d \theta$
is

A) $2 \pi \left( e ^ { 2 } - 1 \right)$
B) $2 \pi \left( e ^ { 2 } + 1 \right)$
C) $4 \pi \left( e ^ { 2 } - 1 \right)$
D) $4 \pi \left( e ^ { 2 } + 1 \right)$
E) $3 \pi \left( e ^ { 2 } - 1 \right)$