Force from the water on the gate

[bcd201115]

Homework Statement

A tainter gate for a lake is 21' tall and 14' wide and the water level is 17 feet from the surface to spillway. What force is the water placing on the gate at the top, middle and bottom? Also when the work is done how much water will flow through the gate if the gate is raised 2 ft, 7ft, and 10ft and how high does the gate have to be raised to maintain a flow rate of 2500cfs. the estimated impoundment of water for the lake is 3 billion gallons.

Homework Equations

Can anyone help me get started on this? I would really appreciate it

The Attempt at a Solution

 

[HallsofIvy]

The pressure due to water at depth h feet, in pounds per square foot, is the weight of a column of water one foot in area and h feet high. Taking the (weight) density of water to be $\delta$ pounds per cubic foot, that is $\delta h$ pounds per square foot. To find the total force on a rectangular gate, of width W and height H, imagine a thin rectangle at depth h, of width W and height dh. Its area is Wdx and the force on it is $\delta hW dh$. Integrate that over from 0 to H.

 

[bcd201115]

So the force at the top would be 0 right? because there's no pressure from the water? and how would i find the area at the middle and bottom?. I'm good on finding pressure but what about the area?