A Plane Lamina with Constant Density $\rho ( x , y ) = 12$

Multiple Choice

A plane lamina with constant density $\rho ( x , y ) = 12$ occupies a region in the xy-plane bounded by a simple closed path C. Its moments of inertia about the axes are $I _ { x } = - \frac { \rho } { 3 } \int _ { C } y ^ { 3 } d x \text { and } I _ { y } = \frac { \rho } { 3 } \int _ { C } x ^ { 3 } d y$ Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0) , (4, 0) , (4, 5) and $( 0,5 )$ .

A) $( - 1,280,1,280 )$
B) $( - 2,880,1,200 )$
C) $( 2,000 , - 1,280 )$
D) $( 2000,1280 )$
E) $( - 2,000,1,280 )$