Josephood7

Evaluate ∫∫S1+x2+y2−−−−−−−−−√dS∫∫S1+x2+y2dS where SS is the helicoid: r(u,v)=ucos(v)i+usin(v)j+vkr(u,v)=ucos⁡(v)i+usin⁡(v)j+vk, with 0≤u≤5,0≤v≤1π

Calculate ∬Sf(x,y,z)dS∬Sf(x,y,z)dS For

∬Sf(x,y,z)dS=

Evaluate the integral with respect to surface area ∫∫T18xdA∫∫T18xdA, where TT is the part of the plane x+y+4z=6x+y+4z=6 in the first octant.

Evaluate the surface integral ∫SF⋅ dS∫SF⋅ dS where F=⟨3x,−4z,4y⟩F=⟨3x,−4z,4y⟩ and SS is the part of the sphere x2+y2+z2=4x2+y2+z2=4 in the first octant, with orientation toward the origin.

Evaluate the surface integral ∫SF⋅ dS∫SF⋅ dS where F=xyi+5x2j+yzk F=xyi+5x2j+yzk  and SS is the surface z=xey, 0≤x≤1,0≤y≤1z=xey, 0≤x≤1,0≤y≤1, with upwards orientation.

Suppose FF is a radial force field, S1S1 is a sphere of radius 22 centered at the origin, and the flux integral ∫∫S1F⋅dS=4∫∫S1F⋅dS=4.
Let S2S2 be a sphere of radius 66 centered at the origin, and consider the flux integral ∫∫S2F⋅dS∫∫S2F⋅dS.

Determine whether the flux of the vector field F⃗ (x,y,z)=zi⃗ F→(x,y,z)=zi→ through each surface is positive, negative, or zero. In each case, the orientation of the surface is indicated by the gray normal vector.