Surface Integrals

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  1. Evaluate the surface integral Sx2yzdS if S is the part of the plane z=1+2x+3y above the rectangle 03×02 .
    17114
  2. Evaluate the surface integral SyzdS if S is the part of the plane x+y+z=1 that lies in the first octant.
    324
  3. Evaluate the surface integral Sx2z2dS if S is the part of the cone z2=x2+y2 that lies between the planes z=1 and z=3 .
    3642π3
  4. Evaluate the surface integral SydS if S is the part of the paraboloid y=x2+z2 that lies inside the cylinder x2+z2=4 .
    π39117+160
  5. Evaluate the surface integral Sx2z+y2zdS if S is the part of the hemisphere x2+y2+z2=4 above the xy-plane .
    16π
  6. Find the flux of F across S if Fxyz=xyyzzx and S is the part of the paraboloid z=4x2y2 that lies above the square 0x1 , 0y1 , and has upward orientation.
    713180
  7. Find the flux of F across S if Fxyz=xzy and S is the part of the sphere x2+y2+z2=4 in the first octant, oriented towards the origin.
    43π
  8. Find the flux of F across S if Fxyz=0yz and S is a positively oriented surface which consists of the paraboloid y=x2+z2 , 0y1 , and the disk x2+z21 , y=1 .
    0
  9. Find the centroid of the hemisphere x2+y2+z2=a2 , z0 .
    00a/2
  10. A fluid has density 870 kg/m3 and flows with velocity v=zi+y2j+x2k , where x , y , and z are measured in meters and the components of v are measured in meters per second. Find the rate of flow outward through the cylinder x2+y2=4 , 0z1 .
    0 kgs