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a) Find the average value f of thefunction \(f(x,y,z)=\sqrt{x^{2}+y^{2}+z^{2}}\) defined atall points of the halfball \(x^{2}+y^{2}+z^{2}<=1\) withz<=0.
b) A capped reservoir is bounded by thesurfaces: \(S1:x^{2}+y^{2}=9\) , \(S2: y+z =5\) , \(S3: z=1\) Calculate the reservoirvolume.
c) Consider three cylinders in the dimensionalspace: \(x^{2}+y^{2}=1\) , \(y^{2}+z^{2}=1\) and \(x^{2}+z^{2}=1\) . Make a sketch ofthe intersection of the three cylinders and find the volume of thesolid bounded by the three cylinders.
Calculate the force and the gradient.
Calculate the volume of the solid bounded by the cylinder z = x^2 and the planes z=2âx,y=0, y=2