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-. 0-’ 1VIATH132O: Volumes by Cylindrical Shells Instructor: Laura Strube 31 August 2015 Turn in for participation credit So(u+iOflc Name and Class ID: 1. Set up and evaluate the integral for the volmue of the solid that results when the region 0 is rotated about the line x 4. Begin bounded by y = + 2x + 3, y = 0, and x by sketching the region. (a) Complete the square to convert y = 2 + 2x + 3 into the form: y —x = a(x — 2 + A:. li) (i (i)aQ 2 —(x (b) Sketch the region and then sketch the solid. 4 4% 1 k ‘C 4. d smrncfr (c) Sketch an approximating shell on the solid. 1.x 1 s&--c c on -tue (d) For this problem, is the width of the shell L x or ‘V S& sk-eh CLôt.,€: (i.e. we 1 (e) Is the height of the shell measured alollg the x or y axis? ‘Write the equation for the height of the shell. Se £ & f-h - aLo v e: h ih j: is iettu rd 7J 0Jo 24243 pejh+= (_X2#243 —, (° 3 _X22X+ (f) Is the radius of the shell measured along the x or y axis? Write the equation for the circumference of the shell. J-tdiLcs IS ,Te LL’t1.S aJ.Onff 1 A : — x) 7me x ‘s x X W C a, 33 J // J1Q 4t& I (ts she/I zx izzdis 4— S. fl7tZLLL4 7C (g) Write the equation for the volume of a single shell. srj.e she/I: Su-rIu tzrz /Oli%rn-e ôí 2ff(4-x( XZ+X3) s;nJecJr/L (h) Write the integral for the volume of the solid and solve. 3 VoIu-rr =r [Your solution should be H] (4_x(xa÷ +3)cJx