Real Functions of Several Variables Examples of Surface by Leif Mejlbro

By Leif Mejlbro

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Extra resources for Real Functions of Several Variables Examples of Surface Integrals Calculus 2c-8

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First find the normal vector of the surface N(u, v). 1) The surface integral F xz 2 dS, where the surface F is given by for 0 ≤ u ≤ 1, 0 ≤ v ≤ 2π. x = r(u, v) = (u cos v, u sin v, hv), 2) The surface integral F z 2 dS, where the surface F is given by √ √ x = r(u, v) = ( u cos v, u sin v, ev ) 3) The surface integral x = r(u, v) = √ 4) The surface integral F ln(2u) ln u ≤v≤ . 2 2 (x2 + y 2 ) dS, where the surface F is given by u cos v, F for 1 ≤ u ≤ 2, √ 3 u sin v, v 2 for 1 ≤ u ≤ 2, 0 ≤ v ≤ u. (x3 + 2z − 3xy) dS, where the surface F is given by x = r(u, v) = (u + v, u2 + v 2 , u3 + v 3 ) for u + v ≤ 0, u2 + v 2 ≤ 5.

Or have you already graduated? P. Moller - Maersk. 3. 2) We get from dS 10 + x2 dx dy = = F 1+ g = (x, 3) that E 1 = 1+ −1 x2 6 Then by the substitution x = dS = F = 1 3 20 3 = 20 3 = 5 3 = = = = = 5 6 0 0 √ 10 + x2 . The surface area is 1 1 −1 − x6 10 + x2 dy 2 2 6 (6 + 10 sinh2 t) · Arsinh( √110 ) Arsinh( √110 ) 0 = 1 (6 + x2 ) dx 10 + x2 dx. 0 √ x , 10 sinh t, t = Arsinh √ 10 Arsinh( √110 ) 0 2 10 + x2 dx = Arsinh( √110 ) 0 g Arsinh( √110 ) √ √ 10 cosh t · 10 cosh t dt (3 + 5 sinh2 t) cosh2 t dt 3 5 (1 + cosh 2t) + sinh2 2t dt 2 4 5 6 + 6 cosh 2t + (cosh 4t − 1) dt 2 {7 + 12 cosh 2t + 5 cosh 4t}dt 5 5 7t + 6 sinh 2t + sinh 4t 6 4 Arsinh( √110 ) 0 Arsinh( √110 ) 5 7t + 12 sinh t 1 + sinh2 t + 5 sinh t 1 + sinh2 t · (1 + 2 sinh2 t) 6 0 1 1 1 2 11 11 11 5 7 ln √ + + 12 · √ · +5· √ · · 1+ 6 10 10 10 10 10 10 10 √ √ 6 + 11 12 √ 6 √ 35 5 1 + 11 3 √ √ · 11 + · 11 = ln + 7 ln + · 11.

1 + 9z 4 dz, we get 1 dS π 2 0 = 2π z3 0 O = 2π 1 √ 1 + 9t dt 4 0 √ π (10 10 − 1). 5. 3 Consider the space curve K given by the parametric description π t ∈ 0, r(t) = 3 cos t − 2 cos3 t, 2 sin3 t, 3 cos t , . 2 1. Show that the curve has a tangent at the points of the curve corresponding to t ∈ 0 , π . 2 2. Show that the curve has a tangent at the point corresponding to t = 0. 3. Find the length of K. The curve K is projected onto the (X, Y )-plane in a curve K ∗ . Let O denote the surface of revolution which is obtained by rotating the curve K ∗ once around the X-axis; and C denotes the cylinder surface which has K∗ as its leading curve and the Z-axis as its direction of generators, and which is lying between the curve K and the plane z = −x.

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