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Math_122_200702_Exam2_solution_key

Math_122_200702_Exam2_solution_key - Exam Two MATH 122...

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Unformatted text preview: Exam Two MATH 122 Winter, 2008 Name Section Pg 1 (24 pts) Pg 2 (24 pts) Pg 3 (24 pts) Pg 4 (28 pts) Pg 5 (5 pts) Total (105 pts) Show all your work on the exam paper, legibly and in detail, to receive full credit. No Calculators. 1. (12 points) Find a deﬁnite integral in terms of the variable x representing the area of the region enclosed by the curvesy : 3 — x, y : x + 1, andy : 4. Note: you do not have to evaluate the integral. ‘6 3 L‘ @ 4 : x+ [ x: 5 : a El. [lb/Maj Ax + £4 —(2<+/)j a’X 2.(12 points) Find a deﬁnite integral in terms of the variable y representing the area of the region enclosed by the curvesy : 3 — x, y : x + 1, andy : 4. Note: you do not have to evaluate the inte ral. g.» c. g gzgac 9 grscwc 153 larK’IL/ —> w 9 WW #a/awgt A: lof6 3. (12 points) Find a deﬁnite integral representing the volume that results when the region enclosed by the curves y : (x — l)2 and y : x —l is revolved about the x-aXis. N t : d t ha e to e al ate the 'nte ral. 7—; , 0e you 0110 v v u 1 g (XII) [X /) 4%— ash-I) L 4. (12 points) Find a deﬁnite integral representing the volume that results when the region enclosed by the curves y : (x — l)2 and y : x —l is revolved about the y-aXis. Note: yo do not have to evaluate the integral. _ 32/29“; l3» 35"“ ix“ @421“ S 20f6 5. (12 points) A spring exerts a force of lOON when it is stretched 20cm beyond is natural length. How much work is required to stretch the spring 80cm beyond its natural length? Note: For this problem, you *DO* have to evaluate the integral to get full credit. an,“ a OOnl T:=K9C m=g_ l 3b» chsl loo ZOCM 90 £30 90 : 526 l W:g 3 {55’de D/ b a by L :- HDO 32—00“: /6/”00 _\ 5—K? _, O :5 2, :‘ Ana/I b..— 2’ :l/éo Nﬂ” 6. (12 points) The vat shown in the accompanying ﬁgure contains water tpth of 2 ) é" meters. Set up the integral for the work required to pump all the water to the too e vat. You do not have to evaluate the integral. Use 9810 N/m3 as the weight density of water. (Hint: you need to ﬁnd the cross-sectional area at varying depths in the vat.) m n1 a/UL ﬂ .ﬁm a zv ()3 W I >< CN B ...-.-x.. -~04*-’>' / X AW 2 ﬁrm) -2>v . (a—x) : 7610 (mg-[43 26x) [s-x) : @810 DX'é 6” [s-x) ¢>< <5 3of6 7. (14 points) Evaluate the indeﬁnite integral. I: lxzexdx =3 eX-KL dX , 56x gfc’ am: 10%”— S”? QX 75’ ~ gap/“34X “D I: ><, 9(le a x L a L 36 x L 1 La 2: a o X, ’— 1 X, 4. x 7L C, 8. (14 points) Evaluate the deﬁnite integral. I Iotan’lxdx j‘éqﬂrfx ﬂ/X : {Leila/I X/X ,I l I [144/11 —” {X /_}_XL dz Liz/*XL \/ I zx /%=&X/X 5x540“ “Tg MW xx 4 _I ,, ‘2 Kéanf/K w—L g? 37044” 76 "Pl ﬂ”) 40f6 9. (5 points) Evaluate the indeﬁnite integral . This can be done by integration by parts, or by a good substitution. Hﬂa I) dx I(1+x2)2 @ __—L_ {— gx 50f6 DO NOT REMOVE THIS PAGE! Some formulas W = IF<x>dx A = I (f(x) — g<x>>dx A = I my) — g<y>>dy V = zjmxy — g(x)2)dx V = 240ml — g(y>2>dy If'(x)g(x)dx = f(x)g(x) — jf(x)g'(x)dx judv = uv — Ivdu >X<*>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<>X<*>X<>X<>X<>X<>X<>X<*>X<>X<>X<>X<>X<>X<*>X<>X<>X<>X<>X<************************ Extra Work Space (If necessary) 60f6 ...
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