Use cylindrical coordinates Evaluate tripleintegralE times d

Use cylindrical coordinates. Evaluate tripleintegral_E times dV, where is enclosed by the planes z = 0 and z = x + Y + 8 and by the cylinders x^2 + y^2 = 4 and x^2 + y^2 = 16. Evaluate the integral by changing to cylindrical coordinates.

Solution

in cylindrical coordinates

x=rcos, y=rsin

x²+y²=r²

x²+y²=4=2²,x²+y²=16=4²

z=0,z=x+y+8

0<=<=2,2<=r<=4 ,0<=z<=rcos +rsin+8

dv =r dz dr d

_E x dV

=_{[0 to 2] [2 to 4] [0 to rcos +rsin+8]} rcos r dz dr d

=_{[0 to 2] [2 to 4] [0 to rcos +rsin+8]} r²cos dz dr d

=_{[0 to 2] [2 to 4][0 to rcos +rsin+8]} r²cos z dr d

=_{[0 to 2] [2 to 4]}r²cos (rcos +rsin+8-0) dr d

=_{[0 to 2] [2 to 4]}r²cos (rcos +rsin+8) dr d

=_{[0 to 2] [2 to 4]} (r³cos² +r³cossin+8r²cos) dr d

=_{[0 to 2][2 to 4]} ((1/4)r⁴cos² +(1/4)r⁴cossin+(8/3)r³cos) d

=_{[0 to 2]}  ((1/4)(4⁴-2⁴)cos² +(1/4)(4⁴-2⁴)cossin+(8/3)(4³-2³)cos) d

=_{[0 to 2]}  (60cos² +60cossin+(448/3)cos) d

=_{[0 to 2]}  (30(1+cos2) +60cossin+(448/3)cos) d

=_{[0 to 2]}  (30+30cos2 +60cossin+(448/3)cos) d

=_{[0 to 2]}  (30 +15sin2 +30sin²+(448/3)sin)

=(30(2) +0+0+0) -(0+0+0+0)

=60

_E x dV =60