227_sample_exam_2[1]

227_sample_exam_2[1] - dy dt = ky M& y where y t is the...

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Math 227 Sample Exam 2 Note: this sample represents the range of questions you should be prepared for. The exam itself will be shorter. 1. (10 pts)Evaluate R 1 0 xe ° 2 x dx: Show all steps. 2. (5 pts each) The region in the °rst quadrant of the xy plane bounded by y = p 4 ° x 2 ; x = 0 , and y = 1 is revolved around the y axis. (a) Sketch the solid region swept out. (b) Use the disk/washer method to set up the integral for the volume swept out. (c) Use the shell method to set up an integral for the volume swept out. (d) Calculate the volume swept out. 3. (10 pts) Find the arclength of the section of the curve y = x 2 between the points ( ° 1 ; 1) and (2 ; 4) : 4. (10 pts) A model for limited population growth is given by the di/erential equation
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Unformatted text preview: dy dt = ky ( M & y ) where y ( t ) is the population at time t , M is the maximum sustainable population, and k is a growth constant. Use separation of variables to &nd the general solution. 5. A thin plate occupies the region in the &rst quadrant of the coordinate plane and is bounded by the curves x = 0 , y = x; and y = p 2 & x 2 : (a) (10 pts) Find the x coordinate of its centroid. (b) (5 pts) Use the Theorem of Pappus to &nd the volume swept out if the plate is revolved around the line x = & 1 . 1...
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