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Unformatted text preview: MATH 138 1. *a.) b) *d) C) *2. a)
b)
3. (I) b) Winter 2007 Submit all problems marked * 011 Friday, January 26 Assignment 3 pages) H Find the volume of the solid generated by revolving about the :r—axis (y
1 l + 11:2.
Find the vohnne of the solid generated by revolving about the y—axis (:17 0) the region
bounded by .T. = O and :r = l — 5/2. [IllN'rz The substitution :6 = tan{) may be useful] 0) the region bounded by m = —1, :L' = 1. y = O and y 2 Use "washers’ to find the volume of the solid of revolution generated by revolving about
the 3:—a.xis ("I = 0) the region bounded by y = see :1", y = 1, .‘L' = l), and :1; : 7r/A1. Find the volume of the solid generated by revolving about the y—axis the region bounded
by x2 + y2 = (1.2, y = b > 0, :L‘ = 0, for b S y g u, i.e., a. spherical ‘cap’. Given that the volume of a certain solid of revolution is given by V 7r/2 l 7
7r/ ((1 +eos:r)g —12)d;{:,
0 describe and sketch the solid. Find the volume V(a) of the solid generated by revolving about the :z:—axis the region
bounded by y = a, y = 61‘. and :1: : 1, where 0 < a < 1. Find 111(1)] i/(a), and explain why this result seems unphysieul
u—v + ‘ I The curve y = is rotated about the yuxis to form a bowl. Find the volume l»""(/i) of water in the bowl if the bowl is ﬁlled to depth ii. If at small hole is drilled in the bottom of the bowl the force of gravity will cause the water (1V . ,
to leak out at a rate ~3— pr(')portional to the square root oi the depth Ii. Show that the all,
(U2, . _
—— which is constant. dt
a ‘water cloek’ used by the Greeks.) water level will drop at a rate (This is the idea behind a Clepsydm, *4. A Jurabolic lens is formed by rotating about the ::I:—axis, the re ‘ion bounded by '12 = .L', and
l . . g .. J
y2 = 2(:r: — ‘2) for 0 g y S 2. Find the volume of glass required to make the lens.
[IilN'i’z Use cylindrical shells] 5. Show that: if you drill a cylindrical hole of radius b through the centre of a sphere of radius a.
where 0 < b < a, then (i) the height: of the hole is h. = ’Zx/ai2 — b2, and (ii) . . . , 7r .
the remannng volume oi the sphere 15 V = Eh".
) MATH 138  Winter 2007 Assignment #3 Page 2 of 3 *6. a) Find the volume V of the triangular pyrami— dal monument shown at left, given that the cross—sections at distance :1: In. from the top are
equrlateral triangles of Side 1 1n. I'e 'Jkﬁfn b) Let be the cross—sectional area at :3, as shown, and show that V = 14an x fr. 7. Identify each of the following improper integrals Type 1 or Type 2, and then determine
whether the integral converges or diverges. 00 ‘00 ,. ~7'r/2 loo 2
8) / €_3$ d1: b) / t dt 0) / see (Mt) *d) / rive—‘1' d3;
1 ] 1 + t2 0 0 *J/w 1 dt f) f1 1 d *)f°° 1 i 1)]001‘” [
e :r 1 r .11: ___7 __ ,‘
_oog+t2 0 11 g 0 (1+3xy/er 1 1 t2
OC 1‘ '00 1 1 l / >00
i) [so eme+ 1 d3} /1 C117 *k) A (jg: 1) /7r C—I Si“ 23: aim
*in) f2 1 (1! *n) /00 —1——— (151: o) foo cos2 0(10 p) foe _.___l____ d1: 0 > U
0 t— 1 i . e :L‘(ln;r.)2 ‘ 0 U + a)“ + 2”) ~ ; A  ‘00
Suggestion: Try to generalize each result (e.g., the result for a.) implies / 6"” dm converges (l for all a 6 R and all k, > 0). "’ cos J133/2 b _ I s111 1' _ . 8. Use inte ration bv :iarts to ex )ress (117 111 terms of (1:1: and hence move that . 7
 7T .00 , in \/1_'
811133
def converges.
ﬂ x/E 9. This problem involves various applications of improper integrals. 00
a) The Gamma function is defined 011 J: > 0 by l“(;:z:) = / tz"le“"dt'. Use integration by
0 parts to show l‘(n. + l) = nl“('n) for all n > U, and hence find a simple way to express [’(n)
for positive integers n. (It will be helpful to find l“(l) and F(‘2) ﬁrst.) *i)) The daily rate at, which people get sick (hiring a flu epidemic in a certain town is given by
M!) = li)()()te""‘ where t is measured in days since the start of the epidemic, and v[t) is in people per day. (i) Sketch a graph of 7‘05), indicating on what day the largest number of new cases occur. MATH 138 — Winter 2007 Assignment #3 Page 3 of 3 (ii) If the total population of the town is 5000, how many people in total will ultimately
avoid becoming ill? [HINTz How can you determine the total number of people who do
become ill for 0 g 1‘. < 00?] c) Find the “volume” of the solid generated by revolving the region bounded by ; = e‘rg/Z
about the y—axis, for 0 g :I: < oo. [HIN'I‘z Use vertical cylindrical shells] ‘ ‘ I v . 2 ‘flf 2
*d) The error l‘unetion oil (:13) is deﬁned In! erl (1;) = e" (If.
7" e (1) Explain how you know that > 1 + :L" for all :1: > 0, and hence l
a” < for all t> 0.
1 +12
Hence prove that
2
0 < e1‘f($) < — arctan :‘L‘ for all a: > 0.
(7? (ii) Use the result of to show that (.‘I‘f(."II) is bounded on :1; > 0. (iii) Use FTCl to ﬁnd eri" and hence Show that erf(;r) is an increasing function for all
1'. and has a point of inflection at :r = 0. Then sketch erf(:z;). 10. Use the Comparison Theorems for improper integrals to determine Whetl’ier each integral
converges or diverges *3.) jidem b) C) ‘ en dt *d) fool.de
1 ‘1 .0 621+t2 6 Int CHALLENGE: 11. 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 1?, which
maximizes V. ...
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