The Moment of Inertia of a Lamina in the Shape $\rho ( x , y ) = y$

Multiple Choice

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by y = sin x, the x-axis, from x = 0 to x =? with area density $\rho ( x , y ) = y$ about the y-axis is

A) $\frac { \pi \left( 2 \pi ^ { 2 } - 3 \right) } { 9 }$
B) $\frac { \pi \left( 2 \pi ^ { 2 } + 3 \right) } { 18 }$
C) $\frac { \pi \left( 2 \pi ^ { 2 } - 3 \right) } { 18 }$
D) $\frac { \pi \left( 2 \pi ^ { 2 } + 3 \right) } { 24 }$
E) $\frac { \pi \left( 2 \pi ^ { 2 } - 3 \right) } { 24 }$