APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the solid obtined by rotting the region bounded between the grphs of f nd g bout the horizontl line y. b The volume of the solid obtined by rotting the region bounded between the grphs of f nd g bout the verticl line. c The surfce re obtined by rotting the grph of f on the intervl bout the -is. d Show tht, on the intervl, the rc length of f is equl to the rc length of g. HINT: You do not ctully need to evlute ny integrls here. + - - - Wsher Method: V Shell Method: V b Shell Method: V Wsher Method: V c S S 4 4 π R r d πy + πr + y d πr + dy 4 y π [ 7 + 5 ] d dy + π y + y dy πrh d π [ 4 + ] d [ π + 4 y ] 4 y dy [ + π + y ] y dy 6 π + + 4 d πy + 4y dy d Since f nd g, the rc length integrl is the sme for both curves: L + y d + 4 d.
Alternte solution: The two prbols re relted by reflection nd trnsltion nd therefore hve the sme rc length.. 6 pts Consider the trpezoidl region with uniform density ρ shown below. Find the moments M y nd M of the region using integrtion. b Find the centroid of the region using your nswers for prt. c Now drw horizontl line t y to divide the trpezoid into tringle nd rectngle. Locte the centroids of the smller regions nd use dditivity of moments to confirm your nswer for prt b. This problem is similr to HW 7.6 #5. The eqution of the line connecting the points, 5 nd, is y + 5. M y M ρf d ρ + 5 d ρ ρ f d ρ + 5 d ρ + 5 d ρ ρ 9 45 + 75 9 ρ ] [ + 5 ρ 9 + 45 7 ρ ρ + 5 d [ 5 + 5 b First find the mss m of the trpezoid, then find the centroid, ȳ. Alterntively m m hb + b ρ 5 + ρ ρ, ȳ My m, M m 7ρ ρ, ] 9ρ 9 ρ 7, 7 ] ρ + 5 d ρ [ + 5 ρ.
c The centroids of the tringle nd rectngle re locted t, nd,, respectively. The msses of the tringle nd rectngle re m 9 ρ nd m 6ρ, respectively. Use dditivity of moments to find the centroid of the trpezoidl region. M y m m ȳ M m m i i m i i 9 ρ + 6ρ 9 ρ 7 m i y i 9 ρ + 6ρ ρ 7 Note tht the centroid of tringle is locted t the intersection of the medins, or of the wy from ech verte to the opposite side.. 6 pts Consider the differentil eqution dy d Solve the differentil eqution. ye. b Find the solution to the differentil eqution tht stisfies the initil condition y. dy d y dy ye }{{} u dud e d }{{} dve d ve y e e d y e e + C y e e + C y ± e e + C b ± + C C 6, which yields the solution y e e + 6. Note tht only the positive squre root mtches the initil vlue. 4. pts Consider the series n where n n Write the prtil frction decomposition of n. b Find simple epression for s n. c Is {s n } monotonic? Justify your nswer. d Is {s n } bounded? If so, find upper nd lower bounds for s n. n + n +. Let the prtil sum s n e Does the given series converge? If so, wht does it converge to? i. i
n n + n + b s n 4 + 5 + 4 6 + + n + n + n + n + s n + n + n + 5 6 n + n + c The n terms re ll positive so s n is n incresing sequence nd therefore monotonic. d s n is bounded below by /4 nd bounded bove by 5 lim s n lim 6 n + n + e The series converges to the limit of the prtil sums, or 5/6. 5. pts For ech of the following, plese nswer True or Flse. Provide brief justifiction of your nswer. 5 6. The sequence { n }, given by n lnn! lnn +! for n,,,... converges. b If lim n then n is convergent. c The series n + n n converges. n d Let n e n nd b n tn n. The series n + b n diverges. n e Suppose you re slicing ten inch long crrot relly thin from the greens end to the tip of the root. If ech slice hs circulr cross section with re f π[r] for ech between nd, nd we mke our cuts t,,..., n so nd n then good pproimtion for the volume of the crrot is f i i. i n! Flse. n lnn! lnn +! ln ln/n + lnn +. Therefore, n +! lim n so the sequence diverges. b Flse. The hrmonic series is counteremple. c True. The series equls the sum of two convergent geometric series n + n with rtios n r nd r, respectively, both less thn. d True. By the Test for Divergence, lim e n + tn n π/ so the series diverges. Alternte Solution The series e n is convergent geometric series with rtio r e <. The series n tn n diverges by the Test for Divergence: n convergent series nd divergent series is divergent. n lim tn n π n. The sum of
e Flse. The correct pproimtion is f i [ i i ] i f i i. i