Question

A nonuniform electric field is given by the expression vector e = ay i + bz j + cx k, where a, b, and c are constants. determine the electric flux (in the +z direction) through a rectangular surface in the xy plane, extending from x = 0 to x = w and from y = 0 to y = h. (use any variable or symbol stated above as necessary.)

Answer 1

The flux through a surface is given by the scalar product of the field strength and the area of the surface. 

As the rectangular surface is in xy plane the flux corresponding to x and y component of field will be zero (parallel to the surface) and the flux through the surface is only due to z component of the field. 
Because flux is the dot product of Electric field and area vector. So, parallel to surface is 0.
So,
The z component of the field is varying with x only and directly proportional to x coordinate. (cx k)

Consider an infinitely thin strip of the surface at a distance x from y axis of thickness dx as in figure. Field at this strip will have magnitude c*x and will be normal to the surface hence the flux through this strip will be given by :