# A3 - 3,x = 0 rotated about the x-axis(e Region bounded by y...

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MATH 138 Winter 2011 Assignment 3 Topics: Volumes by disks, volumes by cylindrical shells. Due: 11 a.m. Friday, January 28. Hand in your solutions to the following problems. 1. Evaluate the following integrals: (a) R 1 0 x 3 - 4 x - 10 x 2 - x - 6 dx (b) R x 2 - 2 x - 1 ( x - 1) 2 ( x 2 +1) dx 2. Make a substitution to express the integrand of R cos x sin 2 x + sin x dx as a rational function and then evaluate the integral. 3. Find the volumes of the following solids generated by rotating the region bounded by the given curves about the speciﬁed line. Specify whether you use the disk method or the cylindrical shells method. (a) Region bounded by y = ln x,x = 0 ,y = 1 ,y = 2 rotated about the y -axis (b) Region bounded by y = 3+2 x - x 2 ,x + y = 3 , rotated about the y-axis (c) Region bounded by y = 1 /x,y = 0 ,x = 1 ,x = 3 , rotated about y = - 1 (d) Region bounded by x = 4 y 2 - y

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Unformatted text preview: 3 ,x = 0 , rotated about the x-axis (e) Region bounded by y = sin x,y = cos x,x = 0 ,x = π 4 , rotated about y = 1 . (f) Region bounded by y = 1 x 2 +3 x +2 ,y = 0 ,x = 0 ,x = 1 , rotated about y-axis 1 4. Find the volume of a torus (doughnut shape) with radius from the center of the hole to the center of the ring R , and radius within the ring r . (See Ex. 63 in Section 6.2 for an illustration.) You may use either method. 5. Find the volume of the following solids: (a) A tetrahedron (pyramid with height h and base an equilateral triangle with side a . (b) A solid with base enclosed by the parabola y = 1-x 2 and the x-axis; cross-sections perpendicular to the y-axis are squares. 2...
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## This note was uploaded on 07/24/2011 for the course MATH 138 taught by Professor Anoymous during the Winter '07 term at Waterloo.

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A3 - 3,x = 0 rotated about the x-axis(e Region bounded by y...

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