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Why must we use integral to find the resultant force?

1. The problem statement, all variables and given/known data
Hydrostatic force on a plane surface ex:
upload_2017-11-13_3-15-19.png


Hydrostatic force on a gate:
upload_2017-11-13_3-12-35.png


2. Relevant equations

3. The attempt at a solution
why cant we just use the formula in the red box above for problem 3.57?
instead I have to use integral
upload_2017-11-13_3-38-14.png

upload_2017-11-13_3-17-44.png

upload_2017-11-13_3-17-14.png

I am confused, how does this gate different from a slanted plane. if I look at the gate from front view, i feel like they are just the same thing. How does
upload_2017-11-13_3-38-24.png
a double integral with respect to area (dhdb) became a single integral with respect to dh?
 

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oh! I can use it but because its the top view, we have to use 5 meter for h_c, which is the depth that affects the pressure.
 
ok, I have one question left, how does
upload_2017-11-13_3-38-24-png.png
a double integral with respect to area (dhdb) became a single integral with respect to dh?
 

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haruspex

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double integral with respect to area (dhdb) became a single integral with respect to dh
Because the width (the b variable in dbdh) is constant.
how does this gate different from a slanted plane.
It doesn't really. Just bear in mind that the gates are vertical, so in the equation in the red box you would put θ=π/2. On the other hand, the area is not WD. To express the answer in terms of W and D you need to make reference to the 15° angle.
With those adjustments, I think you will find the two expressions are the same.
 
Because the width (the b variable in dbdh) is constant.

It doesn't really. Just bear in mind that the gates are vertical, so in the equation in the red box you would put θ=π/2. On the other hand, the area is not WD. To express the answer in terms of W and D you need to make reference to the 15° angle.
With those adjustments, I think you will find the two expressions are the same.
oh yeah, I got it now, thank you!
 

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