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Unformatted text preview: MAT 21BA, Spring 2007, Prof. Opmeer
Thursday June 14, 2007. Final NAME (print in CAPITAL LETTERS, last name ﬁrst): ____ ID#: ____________________________ ._ Instructions: 0 Read each question carefully and answer in the space provided. If the space provided is not enough, please continue on the opposite (blank) side and CLEARLY INDICATE
this. 0 YOU MUST GIVE CLEAR AND REASONABLY COMPLETE ANSWERS TO RE
CEIVE FULL CREDIT. Answers like ’yes’ or ’no’ or ’2’ will be awarded no credit. Proper notation and (mathematical) readability of your answers play a role in deter
mining credit. . Calculators, books, notes or similar things are not allowed. 0 The numbers between brackets refer to the number of points (out of 100) for that
exercise. 1.[30] Compute the following indeﬁnite integrals.
Indicate the substitution that you make ( identity that you use (if you use one).
(a)[6] f(a: + 1)2e‘” d1: if you make one) and state the trigonometric (W61 f m d“ __L__ e H + L
Stu at 1'2, x" _ 3* $L¥~Z3
x = a“ ‘ .
29n\x2\’ Q“\‘”‘\]*C' £3 \ : ~93 '?>~— ~\‘
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293:. (t 2' y) 1.[continued]
(c)[6] f sin3 :1: cos‘ x dx ‘ : s
ngkx (:0qu %“x dx u co x
oh; 2 .—<a,\(\>< dx ’ g ( \ycoSkﬁ) (95“ K 69¢»qu (d)[6] f “#353 da: : LS 9»! dx U: 121M
2 2 \
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7 7: MKB:*§€Q29 \ l.[continued] )(h = Cast 8
(e)[6} f (£11747! d3 (ix: ch, 9 MM} d9 I gegé "an 9 A Er ” S x?“ wL‘ﬁeceernB d9
(fay/$9 2.[12] Evaluate the following improper integrals.
(a) [6] ff” e“ d2: :Qjm Sb 4 $9,”de ‘D'Wo \8 dx '3 “éKkg
: LPN ~éx\b (b)[5] fo°° (Viv; dx 3.[12] Consider the region bounded by the xaxis, the graph y = 3:1:4 and the lines a: = —l and x = 1. Setup, but do not evaluate, integrals which represent the volume of the solid
formed by revolving this region about (a)[3] the zaxis using the disc method.
ﬂog“ %o\id OéUv‘de
\M ~. *wv‘h (b)[3] the yaxis using the shell method“ gag: W“ '
\Io\‘ 93V( ' ‘ﬁ‘ *RWSS (d)[3] the line y = 3 using the disc method.
AC3 NM» Lynda x=l5‘\ 4.[12] Consider the region bounded by the graph y = m2 + 1 and the line y = 5. Setup, but do not evaluate, integrals which represent the volume of the solid formed by revolving this
region about (a) [3] the yaxis using the disc method.
(Ac/g N 60MB Ug‘x‘mCWC (b)[3] the xaxis using the shell method.
9 $‘ €06“ _
\lg n Qw ‘(V‘K’RICXJW‘bS (c)[3] the line y = 5 using the disc method. gbhcl UAVM‘E‘
\lt\ «. “(w 2h * (d)[3] the line y = 5 using the shell method. HCS; énell
V0\ " vah flinch?“ 5.[6] Setup, but do not evaluate, an integral which gives the length of the following curve.
27(t) = t3  6t”. W) = t3 + 6t”, 0 g t g 1. 3.x; 9>t\2£) guw’c L: 8: it] 6.[6] Setup, but do not evaluate, an integral which gives the area of the surface generated
by revolving the curve y = Zﬁ, 1 S x 5 2 about the xaxis. 69‘ : 31¢. cadenﬁfa 7H2] (XJSMH‘)
(a)[6] Setup the area between the m—axis and the graph of the function 2:2  4x — 5 over the interval [—7, 7] in terms of integrals. Do not evaluate. Do not leave absolute value signs in
your ﬁnal answer.  S ‘7
A=S \(quxasB 6x + S \©—~ (xtapsﬁ Ax + 85(X1—4x's\ «ix 1 " (b) [6] Setup the area of the region enclosed by the curves y = 23mm and y = sin 22: with 0 S a: _<_ 27r in terms of integrals. Do not evaluate. Do not leave absolute value signs in your
ﬁnal answer. :r 28km aemx = emZx
Qsinx v <93an ((381 ngxm‘n {133M 2 0
cl («aim @on I \\ :« o CoSX—\ " O
aﬁx7\ X: O‘TIQW x; 013'“ <~3\l\x 1 Q , 8.[10]
(a)[5] Find f(4) if fox2 f(t)dt = zcos ms. at £03) ‘ 61K : Ccmw. r 'mugimrx
z : C(3an r “mint Y
5% (70 ._..__._________.._9x
4‘00: Rf) =r COSQT witlg‘mlﬁ
W
Z .2  9
?Locb in
K22 (b) [5] Use the deﬁnition of the natural logarithm as an integral to show that for all a and b
with b > a > 0 the following holds: lnb—lna 1
“"_“—<. 1<
b b—a a X
6:9?" ,QnX : & €5th
QCK) = /QX\X Q “<7 = ;; I QemW W ‘ (‘n {1‘ ]
QM? Commwds ; so)”,
530034013 mez 91”” dxgoemmw (M at
 : J  29» £00) ~§(a~3 T Q! %(
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