- Find ∫
_{S}xdS where S is the surface which results from the intersection of the cone z = 2 −with the cylinder x^{2}+ y^{2}− 2x = 0. - Now let n be the unit normal to the above surface which has positive z component
and let F=. Find the flux integral ∫
_{S}F ⋅ ndS. - Find ∫
_{S}zdS where S is the surface which results from the intersection of the hemisphere z =with the cylinder x^{2}+ y^{2}− 2x = 0. - In the situation of the above problem, find the flux integral ∫
_{S}F ⋅ ndS where n is the unit normal to the surface which has positive z component and F =. - Let x
^{2}∕a^{2}+ y^{2}∕b^{2}= 1 be an ellipse. Show using Green’s theorem that its area is πab. - A spherical storage tank having radius a is filled with water which weights 62.5 pounds per cubic foot. It is shown later that this implies that the pressure of the water at depth z equals 62.5z. Find the total force acting on this storage tank.
- Let n be the unit normal to the cone z = which has negative z component and let F =be a vector field. Let S be the part of this cone which lies between the planes z = 1 and z = 2.
Find ∫

_{S}F ⋅ ndS. - Let S be the surface z = 9 −x
^{2}−y^{2}for x^{2}+ y^{2}≤ 9. Let n be the unit normal to S which points up. Let F =and find ∫_{S}F ⋅ ndS. - Let S be the surface 3z = 9 −x
^{2}−y^{2}for x^{2}+ y^{2}≤ 9. Let n be the unit normal to S which points up. Let F =and find ∫_{S}F ⋅ ndS. - For F =, S is the part of the cylinder x
^{2}+ y^{2}= 1 between the planes z = 1 and z = 3. Letting n be the unit normal which points away from the z axis, find ∫_{S}F ⋅ ndS. - Let S be the part of the sphere of radius a which lies between the two cones ϕ = and ϕ =. Let F =. Find the flux integral ∫
_{S}F ⋅ ndS. - Let S be the part of a sphere of radius a above the plane z = ,F =and let n be the unit upward normal on S. Find ∫
_{S}F ⋅ ndS. - In the above, problem, let C be the boundary of S oriented counter clockwise as
viewed from high on the z axis. Find ∫
_{C}2xdx + dy + dz. - Let S be the top half of a sphere of radius a centered at 0 and let n be the unit
outward normal. Let F =. Find ∫
_{S}F ⋅ ndS. - Let D be a circle in the plane which has radius 1 and let C be its counter clockwise
boundary. Find ∫
_{C}ydx + xdy. - Let D be a circle in the plane which has radius 1 and let C be its counter clockwise
boundary. Find ∫
_{C}ydx − xdy. - Find ∫
_{C}dx where C is the square curve which goes from→→→→. - Find the line integral ∫
_{C}dx + y^{2}dy where C is the oriented square - Let P=,Q=. Show Q
_{x}−P_{y}= 0. Let D be the unit disk. Compute directly ∫_{C}Pdx + Qdy where C is the counter clockwise circle of radius 1 which bounds the unit disk. Why don’t you get 0 for the line integral? - Let F = . Find ∫
_{C}F⋅dR where C is the curve consisting of line segments,

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