Given #f(x, y)=x^2+y^2-2x#, how do you the volume of the solid bounded by #z=(f(x, y)+f(y,x))/2-5/2, z = +-3?#

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Answer 1 · 2016-10-26T17:06:59

#9pi # cubic units.

The section of this sold by a plane parallel to the xy-plane is the

circle with

center at #(1/2, 1/2, z) #and radius #R(z)= sqrt(z+3/2)#

For integration to find the volume V, choose an element in the form

of a circular disc of thickness #Delta z# and radius R. The faces of

this disc are parallel to the xy-plane.

Now, V = limit #Delta z to 0# of #sum piR^2 Delta z=int piR^2 dz#,

between the limits,from #z = -3# to z = 3..

So, #V = .pi int (z+3/2) dz#, between the limits

#=pi[z^2/2+3/2z],# between #z = -3 and z = 3#

#= 9pi# cubic units.