To get started on the first part - the pressure is the integral of the incremental pressure across the surface of the plate:

where h = depth, w(h) = width of the plate at depth h. You will need to establish a formula for w as a function of h and calculate this integral.

For the second part I suggest a double integral in in dh and dw, together with a moment arm calculation depending on h and w:

where l(h,w) is the moment arm of the point (h,w) from the hinge.