$1.$ A thin rectangular plate lies along the xy plane such that one corner is at the origin and has a density function of $=($x$+2$y$)2.$ It has a length of a units in the x direction and b units in the y direction. a$)$ Find all of the components of the moment of inertia tensor in the coordinate system described. b$)$ If it is rotated around an axis halfway between the positive x axis and the positive z axis, find its moment of inertia about that axis. From this point on$,$ let a$=250$cm and b$=750$cm c$)$ Find its principle moments of inertia d$)$ Find the directions associated with these principle moments of inertia e$)$ This plate is held in place by two supports such that it rotates along the diagonal of the plate in the xy plane. If it is rotated at a frequency of $3\mathrm{.}0$Hz$,$ find the required torque. f$)$ Find the angular momentum in this situation.