M 408D- Final Exam Guide - Comprehensive Notes for the exam ( 25 pages long!)

Show all your work for full credit. The textbook, lecture notes, and elec- tronic devices with symbolic computing capacity are NOT allowed 1. Evaluate cos(x + 2y)dA, whe 2. Evaluate the following double integral Jo Jz 3. Evaluate the double integrals by converting them to polar coordinates. 2心ー昏 4. Evaluate the double integrals by converting them to polar coordinates. 5. Evaluate the triple integrals dxdydz, 2 drdzdy, 3. rdzdrd9. 6. Find the volume between the graph of the function f(r, y) - zy and the rect- angle R = {(z, y)1-2 3and-1 7. Evaluate the following double integrals y 5) 0 z+2 3. 2ded 8. Let S be the triangular region with vertices (0,0), (-1, 1), and (1, 1). (Sketch the region first!) . Find the centroid ofS . If the density at each point in S is proportional to the point's distance from the y-axis, find the mass of S.

Fall 2017 Work all problems on your own paper. Be neat . Show all work. Use graph paper for graphs. You may use your textbook and you This is due Monday, December 4, at the beginning of the elass. I. Determine the area of the region bounded by: y =1 and 2x + 2y-5 2. Set up an integral which represents each volume of revolution indicated. Do not evaluate a. Revolve around the line y-e r notes. Do not consult the internet. Do not use a calculator for integration or differentiation. Giving ce to or receiving ssistance from anyone will be considered cheating- this is a test. Do your own work. a b. Revolve around the y-axis. x=0 y=x evolve around the y-axis. d. Revolve around the y-axis. (2 4

20200 CD page 6 of 6 Fall 2017 Exam #5 exploration! We are in a planet X where the gravity in this planet is g -10 m/sec a mass of 1.0 kg/m. Initially the bucket contains 36 kg of water, but the 9) a leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant spee a rope that has water leak s at a constant rate and finishes draining just as the bucket reaches the 12 m level. How much work is done?