INTEGRAL CALCULUS REFRESHER SET 1. ".
Find the the area area bounded bounded by the the curves curves !" !" # y and y!" y!" # . a. $.% b. $.&&' c. $."%
d. $.(((
)eter*ine the radius radius o+ ,yration o+ the area into the +irst -uadrant -uadrant bounded by the ine y # $/ # ( and the curve !" # (y 0ith resect to y ais. a. 2.'' b. (.33 c. (.$' d. ".("
(. Evau Evauate ate the +oo0 +oo0in in,4 ,4
a. 1%.' 2.
b. 1$.'
c. 11.&
Evauate Evauate the the inte,ra inte,ra o+ "2 d d 6 7% 8 (!"9 (!"9 +ro* +ro* # $ to # 1. 1. a. "1."' b. 15.&% c. (1.13
d. 5.3 d. 22.'"
%.
A ate in the the +or* o+ a araboic se,*ent o+ base 1" * and and a hei,ht o+ 2 * so that that the base base is in the sur+ace o+ the i-uid. Find the +orce on a +ace o+ the ate. a. %&&.5 :N b. (13.& c. 22'.& d. %$"."
&.
Cacuate the de+inite de+inite inte,ra inte,ra o+ !" 8 2 as aroaches aroaches +ro* +ro* 1 to (. a. '26% b. '26( c. '26' d. '265
'.
Cacuat Cacuate e the area o+ the re,ion re,ion to the the e+t o+ araboa araboa # "y!"/ "y!"/ to the ri,ht ri,ht o+ the y ; ais / and bet0een y # 1 and y # (. a. %26' b. %%6& c. %"6( d. %$611
3.
The soid soid has a base 0hich is a circe circe o+ radius radius r. Each cross cross section section erendic erendicuar uar to a +ied +ied dia*eter o+ the circe is an isoscees trian,e 0ith atitude e-ua to one ha+ o+ its base. )eter*ine the vou*e o+ the ,iven soid usin, cross section *ethod. a. 2r!(6( b. 2r6( c. 2r!( d. 2
5.
Find the the area area under under the arch arch o+ y # sin sin bet0een bet0een # $ and # <. a. =1 b. $ c. <
d. "
1$. Evauate this +oo0in, +oo0in, inte,ra4
a. $.133
b. $.(32
c. $.22'
11. Find the Archi*edean Archi*edean sira sira r # > +ro* > # $ and and > # "<. a. 2<6( b. 2
d. $.%1& d. 26(<
1". )eter*ine the the +oo0in, de+inite inte,ra4 inte,ra4
a. 16'
b. "6%
c. (63
d. 165
1(. Let R be the re,ion bounded bounded by the curve curve y # n 6 / the ; ais / and and the ine # e. Find the area o+ this re,ion. a. $.3 b. $.% c. 1.$ d. $ 12. This *ethod is used in cacuatin, cacuatin, vou*e ,enerated by the +unction +unction usin, the cyindrica cyindrica nature o+ the vertica ines o+ revoution. a. ?aus "nd Theore* b. )is: @ethod c. She @ethod d. None
1%. Evauate the +oo0in, +oo0in, inte,ra4 inte,ra4
a."
b. $
c . =1
d. ( 6"
1&. A one horseo0er horseo0er *otor *otor can do %%$ +t b o+ 0or: er second. second. I+ a $.1 h *otor is used to u* 0ater +ro* a +u tan: in the shae o+ a rectan,uar rectan,uar araeied araeied " +t dee/ " +eet 0ide / and & +t on, to a oint % +t above the to o+ the tan:/ ho0 on, 0i it ta:e in ter*s o+ B sec D*e,a indicates unit 0ei,ht o+ 0ater. a. $.13% b. $."&" c. $.(22 d. $.213 1'. Evauate this de+inite de+inite inte,ra4 inte,ra4
a. &%"61%
b. &&161'
c. &2$6"1
13. Find the area bounded bounded by the curves y!" # and and 8 y # ". a. (.& b. 2.$ c. 2.% 15. Find the inte,ra inte,ra o+ sin " d. d. a. ;62 cos " 8 16" sin " 8 C b. ;6" cos " ; 162 sin " 8 C
d. %5361" d. %.$
c. 62 cos " ; 16" sin " 8 C d. ;6" cos " 8 162 sin " 8 C
"$. Find the sur+ace area area ,enerated by rotatin, rotatin, the +irst -uadrant -uadrant ortion o+ the curve curve !" # 1& ; 3y about the y=ais. a. %3.21 b. &2."% c. &1."' d. &&.(3 "1. Evauate the inte,ra inte,ra o+ cos!' ( d 0ith i* i*its its +ro* $ to <6&. a. $.21' b. $.($2 c. $.1%"
d. $."13
"". The *ar,ina cost cost +unction +or a articuar articuar co**odity co**odity is ,iven by C79 # & ; 1'. I+ the cost o+ roducin, " units is ? "%/ +ind the tota cost +unction usin, second +unda*enta theore* o+ cacuus. a. C # (!" 8 1' 8 2' c. C # =(!" 8 1' =2' b. C # (!" 8 1' d. C # (!" ; 1' 8 2' "(. The re,ion bounded bounded by the curve y # tan / the ine # 16( < / and the =ais is revoved revoved about the ais. Find the vou*e o+ the soid ,enerated. a. ".&%% b. ".1%" c. (.$'2 d. ".355 "2. Cacuat Cacuate e this inte,ra4 inte,ra4
a. $.%'"
b. $.33&
c. $.&11
d. $.2'3
"%. A +orce o+ %$$ b is re-uired to co*ress co*ress a srin, 0hose 0hose natura en,th is 1$ in to a en,th en,th o+ 5 in. Find the 0or: done to co*ress the srin, to a en,th o+ 3 in. a. 5$$ in=b b. 1$$$ c. 1%$$ d. 13$$ "&. Evauate this inde+inite inde+inite inte,ra4
a. 162 n 7 = "967 = 29 8 C b. 162 n 7 8 "967 = "9 8 C
c. 162 n 7 8 "967 = 29 8 C d. 162 n 7 = "967 8 "9 8 C
"'. Find the area bounded bounded by y # "67 "67 ; "9/ the ais/ ais/ # 2 / and # %. a. 1.&'' b. 1.1"$ c. 1.(3& d. 1.%15
SITUATIDN 14 The eressions beo0 are the co*onents o+ the acceeration +ro* t # $ to t # 1$ seconds. a # $.3t
ay # " ; $.(t
a # %
"3. Co*ute the the distance aon, co*onent. co*onent. a. "$$ * b. 1((.((
c. 1&5.%$
d. "((.((
"5. Co*ute the the veocity aon, y co*onent. co*onent. a. % *6s b. &.&'
c . 1$
d. 1"
($. Co*ute the *a,nitude *a,nitude o+ the the distance traveed. traveed. a. (1".&% b. (13.%2 c. "22."%
d. "3'.'1
(1. Evauate this inte,ra4
a. "
b. $
c. 16"
d. <62
(". A oint describes describes a ane curve / the co*onents co*onents o+ its veocity at the ti*e ti*e bein, v # %/ v y # "2 ; ("t. Find the distance o+ the oint +ro* its ori,ina osition at the end o+ " seconds. a. 11.5 +t b. 13.5 c. 1%.' d. 1(.& ((. Evauate the inte,ra inte,ra o+ d 6 7(89 0ith the i*its i*its +ro* 1 to &. a. "$6( b. "16% c. 1'65 d. 1"6' (2. Find the s*aest s*aest area bounded by the curv curves es y # sin . sin 76"9 on the +irst -uadrant -uadrant ony. a. 1.% b. 1." c. 1.' d. ".2 SITUATIDN "4 Usin, inte,ration/ (%. Find the *o*ent o+ inertia inertia 0ith resect to ais ais o+ the area bounded in the +irst +irst -uadrant bounded by the araboa y!" # 2// the ine # 1 and the ais. a. 1.1% b. 1.$' c. $.35 d. $.'& (&. Find the *o*ent o+ inertia inertia 0ith resect to y ais ais o+ the area bounded in the +irst +irst -uadrant bounded by the araboa y!" # 2// the ine # 1 and the ais. a. $.15 b. $.(( c. $.2$ d. $.%' ('. Find the inte,ra inte,ra o+ cos 7n d69 +ro* # $ to # (. a. $.%% b. $.'1 c. $.35
d. 1.$'
(3. It is :no0n 0hen +79 # 2 0hen # ". hat is the vaue o+ +79 +79 0hen # $ a. " b. 3 c. & d. $ (5. )eter*ine the area area bounded by the curve y!" y!" # 56% and the ine y # ; ". a. &.3& b. %.'% c. 5.(5 d. 3.'3 2$. Find the area bounded bounded by the curves y # (!" (!" ; " and y # 1 ; 2. a. %%61& b. ("6"' c. (2615
d. 226"(
21. Find the area bounde bounded d by the curve y # 2 6 7!" 8 29/ the ais /the vertica vertica ine # =" and # ". a. " b. $ c. 2 d.< 2". Find the vou*e o+ the +irst +irst octant/ bounded bounded by the sur+ace # 1 and !" # y 8 ". a. $.$' b. $.$% c. $.1$ d. $.$( 2(. Evauate the inte,ra inte,ra o+ d67!" 8 "9 as 0ith the i*its i*its +ro* $ to 1. a. $.'13 b. $.%13 c.$."$( d. $.&2% 22. Evauate Evauate the inte,ra inte,ra4 4
a. n o+ 7 8 "9!"67 ; 19 8 C b. n o+ 7 8 "967 ; 19!" 8 C
c. n o+ 7 ; "9!"67 ; 19 8 C d. n o+ 7 ; "967 ; 19!" 8 C
2%. Find the area o+ o+ the re,ion bounded bounded by r!" # a!" cos "> . a. < b. a c. a!"
d. $
2&. Evauate the inde+inite inde+inite inte,ra4
a. e!" tan 6"
b. ;e! cot 6"
c. e! cot 6"
2'. Evauate the inde+inite inte,ra4 d67a!" d67a!" 8 !"9 !"9 a. arc cosh 6a b. arc tanh 6a c. arc sinh 6a
d. ;e! tan 6" d. Not inte,rabe
23. Find the en,th o+ the arc o+ the the curve "2y # !2 8 23 +ro* # " to # 2. a. 136% b. "163 c. 156( d. 1'6& 25. Find the vou*e o+ the re,ion re,ion bounded by !" # 2y and y # 16" about the y ais. ais. a. &6< b. "<6( c. <6" d. (<6" %$. The +oo0in, *ethods *ethods in evauatin, evauatin, de+inite inte,ras inte,ras EJCE?T4 EJCE?T4 a. Rectan,uar Rue b. Traeoida Rue c. Si*sons Rue
d. A can be used.