$$\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta .$$

Multiple Choice

$$\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta .$$
-Moment of inertia $\mathrm { I } _ { \mathrm { Z } }$ about the $\mathrm { z }$ -axis, where $\mathbf { r } ( \mathrm { t } ) = ( 5 \cos \mathrm { t } ) \mathbf { i } + ( 5 \sin \mathrm { t } ) \mathbf { j } , 0 \leq \mathrm { t } \leq \frac { \pi } { 2 } ; \delta = 2 ( 1 + \sin 2 \mathrm { t } )$

A) $\mathrm { I } _ { \mathrm { Z } } = 10 \left( \frac { \pi + 2 } { 2 } \right)$
B) $\mathrm { I } _ { Z } = 250 \left( \frac { \pi } { 2 } \right)$
C) $\mathrm { I } _ { \mathrm { Z } } = 0$
D) $\mathrm { I } _ { \mathrm { Z } } = 250 \left( \frac { \pi + 2 } { 2 } \right)$