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Firnl. 'foJI. Mulh 2G2 (Fn.11 202 1) C1-1.lc11l11s JI ( 1// ll l J• J

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your answers n11d show your worl< t.o gl'I. full nPdil 1':ud1 part. (J I 1•11d1 pr,>l ,11•111 h ., r!I f1. 1

points. Plt'fLS<' pay 11t1.e11t io11 to wlH'LliN or 1101. yo11 1H :1·d t.fl 1•v1 d1mt ,. I h1• 111 1 , • .,, , 11 111 1 - \

1. Find the ti.rc·n hr.tween the cllJ'VC'H: JI = c·os ;: 1u1d 7/ - r, - r11 .r w}wr1 • (1 f

\:>t, t t-<J rJ- ti-~ (n) First, set up thr. i111.egrnl (do not Pvull ,al<~ '

( [s-w,/\ - e,us i )olX = l ( ("::. - ?.. c•>..,. JX a +oq') - b,,d-.,_ I 0

(b) Now integrate to find th[ exac~SJtsw r ,, '5 j. - 1.s•r. '/. \

0

-------

,r

2. Find the volume of the solid obtained by rotating the region bound d by y = x , JI = I ,

x > 0 about the x-axis. - c:---

(a) Set up the integral using the disk/washer method.

( ' l +~ ~-;,,~s 11 fl~ fJ.l ) Jx , ---1

0 ,.a.. ,..).~, :;: \ 'ii v~.1--~ i \ cl~ 0

cJ ('

0

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3. A force of 20 N iH roquircd to strot.ch n Mpri11g '.t rnuL(1nJ fror11 iLH 1mt11ml l1m 11, tl1. I low rn1wl1 work is do110 in stretching it from !) t.o 22 mcl.1:rH (frorn lt.H nut.11rnl I •11gtl1)'f

(n) Find tho ::;pring comitnnt. k.

10N - k(zM) F ~ K X

(b) 1• iud 0 a tly how mu h work i::i required to 1,t,reLd1 the Hpring.

c:: s l21~~ :: (1.01s~Nl

4. Th lincm den ity of a thin rod is 5+ c s(2x) kg/m, wher , 0 s x s 1.

(a) S t, up the integral to find th - mass of th

t ( l£--1 w; 1 )( J X ) o r Jt

(b) Evaluat th int gral .

St+ ~ 51 (\1 )(

Cy@

rod.

or U) \S J

_ ___ ,

5. The bas of a solid is the area b twe -n y = 42 - x2 and y = 0. Cross sections parallel to the

y- a:. is ar - quar s. Set up the int gra1 to find its volume ( do not integrate) .

• 6. Find the arc~length of the curve y = Gt':~.r from :r = 0 to ,l.

(a) Write down the appropriate integral .

(b) Approxnnate this integral to two decimal places.

7. Find the center of mass of the region bounded by y = x4, y = -I , 0 ~ :r _ l with <lcnsiLy

• (]' = 60.

(a) Find the mass of the region .

(b) Set up the integral to find x (do not evaluate). , J__ \ 60 )( Ci --1 \ J X 72- 0

---

AC!Rf kJ. M{ or Y.

( c) Set up the integral to find fj ( do not evaluate).

\ \_ ,--

• 2 7'1.

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8. Find the surface area of y = 3el: (0 ::; x ::; 6) rotat,ed about the y- axis.

(a) Write down an appropriate integral (do not evaluate yet) .

Vi

(b) (Use your calculator) to approximate this integral to two decimal places . ( ~ '

-1\ wr~~ e~ (Jr;. ,f\(.Cl,{CL.J... t,Jc.J,' I

9. An aquarium is 4 feet tall and has a 8 foot by 8 foot square base. The aquarium is filled to the top with orange juice which has a weight density of 63 pounds per cubic foot .

(a) Set up the integral to find the hydrostatic force on any one side of the aquarium .

~ ~ o4 c~dK ) - \ ~(517-.

( c) Find the hydrostatic force on the bottom of t½_ aquarium.

61., ~ . 75 . ~ t ~ b l 2 '6