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18.022 Recitation Quiz (with solutions) 01 December 2014 1. Let S be the surface defined by z= 1 for z ≥ 1. x2 + y2 (a) Sketch the graph of this surface. (b) Determine whether the volume of the region bounded by S and the plane z = 1 is finite or infinite. (c) Determine whether the surface area of S is finite or infinite. Solution. (a) See the graph below (cut off at z = 10). R 1R 2 R 1R 2 R1 (b) The volume is given by 0 0 π(r−2 − 1)r dθ dr = 0 0 π(r−1 − r) dθ dr. This is infinite since 0 R1 is infinite and 0 r dr is finite. (c) The surface area of S is given by Z 0 1 Z 2π 0 Z q 2 2 1 + fx + f y r dθ dr = 1 Z 2π 0 Z = 0 1 Z 2π 0 0 1 Z = 2π dr r q 1 + x2 /(x2 + y2 )4 + y2 /(x2 + y2 )4 r dθ dr √ 1 + r−6 r dθ dr √ r2 + r−4 dr. 0 This integral is infinite because the second term dominates, and rigorously, we can drop the first term: Z 2π 0 1 √ 1 Z r2 + r−4 dr ≥ 2π 0 R1 dr 0 r2 = +∞. To prove this √ r−4 dr = +∞.