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M138.W09.assmt3 - MATH 138 Assignment 3 g3 pages Winter...

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Unformatted text preview: MATH 138 Assignment 3 g3 pages) Winter 2009 2. a) b) *3. a) Submit all problems marked * on Friday, February 1 Find the volume of the solid generated by revolving about the x-axis (y = 0) the region 1 + 1'27 Find the volume of the solid generated by revolving about the y-axis (a: = 0) the region bounded by :1: = 0 and a: = 1 — ya. bounded by a; = -—1, :1: = 1, y_ = 0 and y = [IVIINTz Use the substitution :1; = tend] Use ‘washers’ to find the volume of the solid of revolution generated by revolving about the z~axis (y = 0) the region bounded by y 2 sec :3, y = 1, a: = 0, and x = 7r/4. Find the volume of the solid generated by revolving about the y-axis the region bounded by 3:2 +3;2 = (22, y = b > 0,11: = 0, for b S y g a, i.e., a spherical ‘cap’. Given that the volume of a certain solid of revolution is given by 7r/2 - V: / ((1+cosa:)2— 12)da:, 0 describe and sketch the solid. Explain your reasoning. Find the volume V((1.) of the solid generated by revolving about the y—axis the region bounded by y = '1, y = lnrc, and a: = a, where 0 < a. < 1. Find li1(1)1 V(a), and explain why this result seems unphysical. a-—* + The curve y = 3:4 is rotated about the y-axis to form a bowl. Find the volume V(h) of water in the bowl if the bowl is filled to depth h. b) If a small hole is drilled in the bottom of the bowl, the force of gravity will cause the water dV . to leak out at a rate E proportional to the square root of the depth h. Show that the ' dh water level will drop at a rate E which is constant. (This is the idea behind a Clepsydra, a ‘Water clock’ used by the Greeks.) *4. A parabolic lens is formed by rotating about the z-axis, the region bounded by y2 = 2:, and 3/2 = 2(33 — 2) for 0 S y S 2. Find the volume of glass required to make the lens. [HINTz Use cylindrical shells] 5. Show that, if you drill a cylindrical hole of radius b vertically through the centre of a sphere of radius a, where 0 < b < (1., then (i) the height of the hole is h = 2\/a2 —- b2, and . 77' ~ . . (ii) the remaining volume of the sphere 13 V = Eh“, 1.e. V depends only on the height h. [HINTz You can use cylindrical shells (tin cans) OR washers to find V] MATH 138 — Winter 2009 Assignment #3 Page 2 01" 3 6. Cavalier’s Principle states that if all the planes y = constant give equal cross—sectional areas for two solids 31 and Sg, then they have equal volume. Use this principle to find the volume of the skewed cylinder shown at right. 7. Identify each of the following improper integrals as Type 1 or Type 2, and then determine whether the integral converges or diverges. 00 ~00 l -7r/2 co 2 a) / 6‘3”” dzr b)/ 2alt c)/ seefldO *d)/ {BC—I dac . 1 7 1 1 ‘H 0 0 °° 1 1 °° 1 °° lnt *3 dt f 1 d , —— * — 1 c) /_009+t2 )/0 J: 112: :r g)/0 (1+3$)3/2d:c 10/] t2 dt *1) /°° ‘31 d2: 1) foidx *k)/1“—“’-"czm 1) fig-ranges .—ooem+1 1 x/E 0 V5 W 2 1 DO 1 oo 00 1 * _ dt d * 2 10 >1: / — ' In) ,/0 t— 1 11) f6 31:(1n:r)2 a: 0) _/0 cos 6( p) 0 (:r + a)(:r + 2a) d513, a > 0 oo Suggestion: Try to generalize each result (e.g., the result for a.) implies / Ends converges ‘ a for all a E R and all k > D). b - b ‘ . SID a: , cos :1: 8. a) Use Integration by parts to express ~77 da: 111 terms of 3/2 (1x , and hence prove V (I: III 7r 1r 0? (2032: $3” dzr. °° sin .r . . that / —— dm converges by applying a Cornparlson Theorem to / Tl' fl 1r 23—‘30 *b) Some people believe that if / f (x) drr converges, then lim f (2:) = 0‘. Show that they are 00 wrong by proving that / sin(a;2) dcc converges even though lim sin(x2) does not exist. fl :r—'oo 2 [IIINT2 Try the substitution u = :L‘ ,anduseyourresultf'roma).] 9. This problem involves various applications of improper integrals. 00 a) The Gamma function is defined on at > 0 by F(:B) = / tx'le‘ldt. Use integration by 0 parts to show I‘(n + 1) = n1"(n) for all-n > 0, and hence find a simple way to express PM) for positive integers n. (It will be helpful to find F(1) and [‘(2) first.) MATH 138 - Winter 2009 Assignment #3 Page 3 of 3 11. *b) The daily rate at which people get sick during a flu epidemic in a certain town is given by r(t) = 10mm“ where t is measured in days since the start of the epidemic, and 'r'(t) is in people per day. (i) Sketch a graph of r(t), indicating on what day the largest number of new cases occur. (ii) If the total population of the town is 5000, how many people in total will ultimately avoid becoming ill? [HINT2 How can you determine the total number of people who do become ill for 0 S t < 00?] c) Find the “volume” of the solid generated by revolving the region bounded by y = 6““2/2 about the y—axis, for 0 S x < oo. [H1NT: Use vertical cylindrical shells] *d) The error function erf (:3) is defined by erf (at) = % / l (3"2 dt. 0 (i) Explain how you know that e” > 1 + 1‘ for all :1: > 0, and hence L2 1 e‘ <1+t2 forall t>0. Hence prove that 2 0 < erf(rr) < E aretan :1: for all a: > 0. (ii) Use the result of (i) to show that erf(:1:) is bounded on a: > 0. (iii) Use FTCI to find erf/(x), and hence show that erf(:z:) is an increasing function for all 11:, and has a point of inflection at a: = 0. Then sketch erf(a:). 10. Use the Comparison Theorems for improper integrals to determine whether each integral converges or diverges * 00—; 00—; d- * [no 6‘ dt d w—l—zt a)/1 $6+1dx b)/1 45r2—1 a, c) 0 e2‘+t2‘ )8 lnt( CHALLENGE: (Hand in to your instructor in class on Friday, if requested.) A martini glass has the shape shown, and is filled to the top with wine. A spherical ball of radius R is gently lowered into the glass, causing it to overflow. Find the volume V of fluid that overflows, and determine the value of R which maximizes V. ...
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