Question

6. (a) (1 marks) Sketch the region bounded by the curves y = sin x, y = x+1, x = 0 and x = - 27. (b) (3 marks) Use the method of cylindrical shells to set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region about the line x = 27. (c) (3 marks) Use the method of washers to set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region about the line y = -2.

Answer 1

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ANSWER – Case(A)Region bounded by curve y = sin(x), y = x+1,x = 0 & x = 2n is given below. 7.5 A B 5 - y = x+1 x = 0 X = 277 2.5+ A y = sin(x) с -10 -7.5 -5 -2.5 8 O 2.5 5 7.5 OABC is the required bounded region. Case(B)Now, Volume of solid of revolution about x = a line by Cylindrical method is given by PV = S 21(a – x)(height) dx here, a = 27 & height = (x + 1) – sin(x) so from egn(1), we get (1) 277 PV= [ 27(27 – x) [x +1- sin(e)] de 0 (2) Case(C) Now, we know Volume of solid of revolution about y = a line is given by =V = [ Te[(outer radius)2 – (inner radius)2]dx here, outer radius = (x + 1) - (-2) = x + 3 & inner radius = sin(x) -(-2) = sin(x) + 2 hence, from eqn(2) 27 BV= $ [(x+3)2 – (sin(2) +2}]dx #AN. f'te