Hydrostatic Pressure Load on a Parabolic Dam Floodgate (Civil & Environmental Engineering)

A water resources and structural civil engineer at the UNLV Howard R. Hughes College of Engineering is analyzing the structural integrity of a submerged irrigation canal floodgate near Lake Mead.

The vertical floodgate is modeled as a planar general region $R$ in the $xy$ -plane bounded on the left by the parabolic retaining edge $x = y^2$ and on the right by the vertical concrete support wall $x = 4$ , with spatial dimensions measured in meters ( $0 \le y \le 2$ ):

R = \{(x, y) \in \mathbb{R}^2 \mid 0 \le y \le 2, \, y^2 \le x \le 4\}

The water surface reaches the crest of the floodgate at $y = 2\text{ m}$ . Hydrostatic fluid pressure $P(x, y)$ increases linearly with submersion depth $(2 - y)$ according to the hydrostatic law:

P(x, y) = \rho g (2 - y) \quad (\text{N/m}^2)

where

\rho = 1000\text{ kg/m}^3

is fresh water density and

g = 9.81\text{ m/s}^2

is gravitational acceleration.

To select high-capacity hydraulic actuator rams capable of opening the gate during an emergency flash flood event, the engineer must:

Identify whether region $R$ is naturally configured as a Type I or Type II general planar region, stating the corresponding integration boundary inequalities.
Formulate the total resultant hydrostatic force $F_R = \iint_R P(x, y) \, dA$ as an iterated double integral.
Evaluate the total horizontal resultant force $F_R$ acting against the face of the floodgate in terms of $\rho g$ , and calculate the numerical load in kilonewtons ( $\text{kN}$ ).