Hydrostatic force acting on a hemispherical impression

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Consider vertical wall which contains a hemispherical belief arsenic shown. Water is filled only upto nan centre of hemispherical impression, i.e. to a extent of L. It is required to cognize nan horizontal unit acting connected nan hemispherical impression, which is successful interaction pinch water.


For doing this, I return a fluid constituent astir nan belief which is fundamentally a cuboid wrong which location is simply a 1/4 sphere impression.

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In my understanding, nan horizontal unit should beryllium adjacent to nan unit connected nan beforehand vertical look of this "cuboid", since nan "cuboidal" fluid constituent has to beryllium successful equilibrium.

So to find that unit I return astatine a extent h, an elemental area dA= dh. 2r. The unit acting connected this constituent will be

$$\delta F= (P_{atm} + \rho gh).2r dh$$

The full unit connected nan beforehand vertical area of nan "cuboid" will be,

$$F= \int_{L-r}^{L}(P_{atm} + \rho gh).2r dh$$

I person been utilizing this method to lick immoderate problems but I'm not getting correct answers. Which makes maine deliberation thing is incorrect pinch this method.

What precisely is incorrect successful this method?