- Find a parametrization for the intersection of the planes 4x+2y +4z = 3 and 6x − 2y = −1.
- Find a parametrization for the intersection of the plane 3x + y + z = 1 and
the circular cylinder x
^{2}+ y^{2}= 1. - Find a parametrization for the intersection of the plane 3x + 2y + 4z = 4 and
the elliptic cylinder x
^{2}+ 4z^{2}= 16. - Find a parametrization for the straight line joining and.
- Find a parametrization for the intersection of the surfaces 4y + 3z = 3x
^{2}+ 2 and 3y + 2z = −x + 3. - Find the area of S if S is the part of the circular cylinder x
^{2}+ y^{2}= 4 which lies between z = 0 and z = 2 + y. - Find the area of S if S is the part of the cone x
^{2}+ y^{2}= 16z^{2}between z = 0 and z = h. - Parametrizing the cylinder x
^{2}+ y^{2}= a^{2}by x = acosv,y = asinv,z = u, show that the area element is dA = adudv - Find the area enclosed by the limacon r = 2 + cosθ.
- Find the surface area of the paraboloid z = hbetween z = 0 and z = h. Take a limit of this area as h decreases to 0.
- Evaluate ∫
_{S}dA where S is the part of the plane 4x + y + 3z = 12 which is in the first octant. - Evaluate ∫
_{S}dA where S is the part of the cylinder x^{2}+y^{2}= 9 between z = 0 and z = h. - Evaluate ∫
_{S}dA where S is the hemisphere x^{2}+ y^{2}+ z^{2}= 4 between x = 0 and x = 2. - For ∈×,let
Find the area of f

. Hint: Check whether f_{θ}⋅ f_{α}= 0. This might make the computations reasonable. - For ∈×, let
where α is such that x =

^{T}. Find ∫_{f([0,2π]×[0,2π]) }hdA. Hint: Check whether f_{θ}⋅ f_{α}= 0. This might make the computations reasonable. - For ∈×, let
where θ is such that x =

^{T}. Find ∫_{f([0,2π]×[0,2π]) }hdA. Hint: Check whether f_{θ}⋅ f_{α}= 0. This might make the computations reasonable. - In spherical coordinates, ϕ = c,ρ ∈determines a cone. Find the area of this cone.
- Let F =and let S be the curved surface which comes from the intersection of the plane z = x with the paraboloid z = x
^{2}+ y^{2}. Find an iterated integral for the flux integral ∫_{S}F ⋅ ndS where n is the field of unit normals which has negative z component. - Let F =and let S denote the surface which consists of the part of the sphere x
^{2}+ y^{2}+ z^{2}= 9 which lies between the planes z = 1 and z = 2. Find ∫_{S}F ⋅ ndS where n is the unit normal to this surface which has positive z component. - In the situation of the above problem change the vector field to F =and do the same problem.
- Show that for a sphere of radius a parameterized with spherical coordinates so
that
the increment of surface area is a

^{2}sinϕdθdϕ. Use to show that the area of a sphere of radius a is 4πa^{2}.

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