# Question 1 Sketch Following Quarter Cylinder R Need Mark Things Sketch Throughout Assignam Q29432011

Please only SOLVE QUESTION 2 quetion 1 is provided forrefrenceThanks

Question 1 (a) Sketch the following quarter-cylinder in R You will need to mark more things on this sketch throughout the assignament, so make it about half a page in sie (b) You already know the volume of a quarter cylinder. Calculate the volume of V. b evaluating the following nested triple integral (which is being given to you) Volume d: dy dz (c) ) Re-write V in set notation using cylindrical coordinates, with polar angle0 taking values in [-1,0]. (B) Also write the triple integral that gives the volume of V in cylindrical coordinates iii) Caleulate the volume of V using cylindrical coordinates Question 2. The region V described in Question 1 is bounded by a surface made of five pinces, S, S. S. S, and S . On the vertical curved surface S1: S is a quarter-circle in the plane0 . Sa is a quarter-circle in the plane 1 S, is a square in the plane y .Ss is a sqaare in the plane -0. Mark Si, Sa,Ss.Sa and Ss on your diagram fron Question 1 We now define a vector field B)-4+2, and continue to consider V and its boundary surface S USUS,US,US (a)Explain wley JJs, B-s- b) Evaluate B dS. (e) Evaluate JIB ds. (d) Evaluate JJs, B-ds. (e) Caleulate V-B and hence write dowm Jy -BaV using Question 1 (0) Using the Divergence Theoren, nd Js, B-dS (g) Let C be the boundary curve of S (only) (i) On your diagram from Question 1, mark your chosen orientation on C (ii) Using Sukrs, Thoma, find the locp integral , dy, Note 1: You do not need to parametrise C to do this question Note 2 Because S lies in the plane1, it may be easily described using plane polar coordinates and the correct area elemet ds-pdp Note 3: In order to apply Stokes thoorem, we may choose the open surface to be S itself, with the loop as its boundary Show transcribed image text

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