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MAT1322 Solution to Final Examination Fall 2012 1 SOLUTION TO FINAL EXAMMAT 1322D, Fall 2012 Total = 53 marks 1.(4 marks) Determine whether each of the following improper integrals is convergent or divergent. If it is convergent, find its value; if it is divergent, use appropriate method to justify your conclusion: (a) 01(1)(3)dxxx∞++∫. . (b) 23012dxxx+∫. 11112.(4 marks) Find the volume of a solid whose base is the region on the x-yplane bounded by the x-axis and the graph of the function y= 2(1 -x2). The cross sections of the solid perpendicular to the x-axis are semicircles with diameter on the x-yplane. Find the volume of this solid. 3. (4 marks) Find the length of the arc y= 211ln24xx-, 1 ≤x≤2.
MAT1322 Solution to Final Examination Fall 2012 2 211216xx-+. 1 + (y')22111++=+= 4. (5 marks) A monument of the shape of a truncated circular cone is built with stone of density 5000 kg / m3. The radius of the bottom of the monument is 3 meters, the radius of the top is 1 meter, and the height of the monument is 20 meters. (Recall that the radius of the cross section at height his 3 -h/ 10 meters). Find the work (in Joules) needed to lift the stone from the ground level to build the monument. (Use g= 9.8 m/sec2)