1.
Volume below the function z=f(x,y) and above the region R is given by R∬f(x,y)dA
V=R∬f(x,y)dA=8∫103∫4(2x+y)dydx=8∫10[2xy+21y2]34dx=
=8∫10(8x+8)−(6x+29)dx=8∫10(2x+27)dx=[x2+27x]810=
=(100+35)−(64+28)=43
Answer: 43.
2.
Using the same formula for z=x2,R={(x,y):0≤x≤6,0≤y≤6}
V=R∬x2dA=0∫60∫6x2dydx=0∫6[x2y]06dx=0∫66x2dx=[2x3]06=432
Answer: 432.
3.
Using the same formula for z=3−x,R={(x,y):x2+y2=9}
V=R∬(3−x)dA=−3∫3−9−y2∫9−y2(3−x)dxdy=−3∫3[3x−21x2]−9−y29−y2dy=
−3∫369−y2dy=6∗(π32)/2=27π
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