Why must we use integral to find the resultant force?

[EastWindBreaks]

Homework Statement

Hydrostatic force on a plane surface ex:

Hydrostatic force on a gate:

Homework Equations

The Attempt at a Solution

why can't we just use the formula in the red box above for problem 3.57?
instead I have to use integral

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

a double integral with respect to area (dhdb) became a single integral with respect to dh?

 

[EastWindBreaks]

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.

 

[EastWindBreaks]

ok, I have one question left, how does

a double integral with respect to area (dhdb) became a single integral with respect to dh?

 

[haruspex]

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.

 

[EastWindBreaks]

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!