Question

consider the region E, which is under the surface z=8-(x^2+y^2) and above the region R in...

consider the region E, which is under the surface z=8-(x^2+y^2) and above the region R in the xy-plane bounded by x^2+y^2=4.

a) sketch the solid region E and the shadow it casts in the xy-plane
b) find the mass of E if the density is given by δ(x,y,z)=z
Science   Advanced-Physics   
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Answer #1

Therefore E is a cylinder with height 8 units and radius 2 units.

the shadow it casts on the x-y plane is a circle of radius 2 with center at the origin.

density = mass/volume

therefore mass = volume x density

The volume of E is =

given that the density is a function of z, to find the mass we consider a small section of the cylinder with height dz and integrate with respect to z from 0 to z=4

mass of the section

total mass is therefore

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