Transcription of APPM 1360 Exam 2 Spring 2017 - University of Colorado …
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APPM 1360 Exam 2 Spring 2017 - University of Colorado …
APPM 1360 Exam 2 Spring 20171. (10 pts) Given the differential equationdSdr+2rS= 0with initial valueS(2) = 3, andr,S >0, solveforSand :dSdr+2rS= 0dSdr= 2rS dSS= 2rdrlnS= 2 lnr+C(sincer,S >0)S=e 2 lnr+C=Ae 2 lnr=Ar 2 Use the initial valueS(2) = 3to find that3 =A(2) 2 A= 12r 22. Parts (a) and (b) are not related.(a) (13 pts) The curvey= cosh(x),0 x 2, is rotated about they-axis. Set up an integral tofind the area of the generated surface, then evaluate the integral.(Hint:Recall thatcosh2(x) sinh2(x) = 1.)Solution:y= coshx, y = sinhxS= ba2 rds= ba2 r 1 + (y )2dx= 202 x 1 + sinh2xdx= 202 x u=xdu=dxcoshxdx dv=coshx dxv=sinhx= 2 ([xsinhx]20 20sinhxdx)= 2 ([xsinhx coshx]20)= 2 (2 sinh 2 cosh 2 (0 cosh 0))=2 (2 sinh 2 cosh 2 + 1)(b)(18 pts) The semicircular arc shown at right extends frompoint( 4,3)to point(5,0).
APPM 1360 Exam 2 Spring 2017 1.(10 pts) Given the differential equation dS dr + 2 r S= 0 with initial value S(2) = 3, and r;S>0, solve for Sand simplify.
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