Consider the $4-$noded quadrilateral element shown in $(x,y)$ coordinates with the dimensions in cm $.$ It is transformed to a square of side $2$ cm in the $(,)$ system with its origin at the center of the square. All integrations need to be done by hand and results must be in the form of improper fractions.

$($a$)$ Obtain the Jacobian matrix.

$($b$)$ Determine the Jacobian.

$($c$)$ Determine the area of the quadrilateral by integrating in the $(,)$ system.

$($d$)$ Determine the centroid of the quadrilateral by integrating in the $(,)$ system.

$($e$)$ Determine $I_x$ of the quadrilateral by integrating in the $(,)$ system.

$($f$)$ Determine $I_y$ of the quadrilateral by

integrating in the $(,)$ system.

Please solve with numerical values... thank you