Chapter 8.3, Problem 18E

Suppose that a plate is immersed vertically in a fluid with density ρ and the width of the plate is w(x) at a depth of x meters beneath the surface of the fluid. If the top of the plate is at depth a and the bottom is at depth b, show that the hydrostatic force on one side of the plate is

$F={\displaystyle {\int }_a^b\rho gxw\left(x\right)dx}$