Solving Hydrostatic Forces with Integral or Volume/Weight?

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I am actually trying to solve a problem from fluid mechanics. The especifications of the problem are on the attachment.

I can solve the problem easily by integration, but I'm not so sure if there is a vertical component of a force in the "dam" acting upward. There is a vertical component of the force acting upward?
If there is, there is another method to solve the problem without integration, using properties such as volume, specific weight... to find that vertical component?
 

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I'm not sure what the various parts of your drawing represent. However, in general, pressure always acts normal to the surface. So if the normal vector of your surface at a given point has a vertical component then the pressure force will also have a vertical component. The easiest way to do something like this is to write an expression for the normal vector as a function of x (or y) and then use that in the integration.