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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 xaxis (y = 0) the region
1 + 1'27
Find the volume of the solid generated by revolving about the yaxis (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 yaxis 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 yaxis 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 zaxis, 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) figranges
.—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 alln > 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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