Find the mass of the lamina in the shape of the portion of the plane with equation 4x + 8y + z = 8 in the first octant if the area density at any point (x, y, z) on the plane is δ(x, y, z) = 6x + 12y + z g/cm^2
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Expert's answer
2022-05-24T09:56:53-0400
ANSWER The mass of the lamina is 96g .
EXPLANATION
The mass of the part of the plane S is calculated using surface integral according to the formula m=∬Sδ(x,y,z)dS , where S={(x,y,z):z=8−4x−8y,0≤x≤2,0≤y≤1−2x} . S has a one-to-one projection onto the domain D in the xy− plane:
D={(x,y):0≤x≤2,0≤y≤1−2x} . The surface area element on S is given by dS=1+zx2+zy2dxdy=1+16+64dxdy=9dxdy . The surface integral of δ(x,y,z) over S can be expressed as a double integral over the domain D
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