Question 5 (30 marks, C4) Let I be the the intersection of the cylinder x² + y² = 4 with the plane x + y + z = 0, and le

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Question 5 30 Marks C4 Let I Be The The Intersection Of The Cylinder X Y 4 With The Plane X Y Z 0 And Le 1 (62 KiB) Viewed 29 times

Question 5 (30 marks, C4) Let I be the the intersection of the cylinder x² + y² = 4 with the plane x + y + z = 0, and let R be the part of the plane x+y+z= that is enclosed inside the cylinder x² + y² = 4. (a) Find a continuously differentiable function : [0, 2π] → R³ that parametrizes I. (b) Evaluate the integral [(y² - x²)ds. (c) Find a continuously differentiable mapping r : D → R³, with D a Jordan domain in R², that parametrizes the surface R. [4] (d) Find the surface area of R. (e) Evaluate the surface integral (2²+ y² + z²)do. R (f) Let F: R³ R³ be the vector field F(x, y, z) = (e²² + y² + x² Use Stokes' formula to evaluate R +y, ex² + y² +₂² curl F. do. -x, ex² + y² +₂² -2). + [3] [6] [5] [5] [7]