CHAPTE TWO MULTIPLE INTEGAL Aft complting ths tutoils, stunts shoul b bl to: vlut th oubl intgl ov th givn ctngul gion fin th volum of th soli boun b th plns fin th of th gion boun b th cuvs ug oubl intgl fin th volum of th soli boun b th gphs ug oubl intgl fin th of th gion b ug oubl intgl in pol cooints chng th intgn fom Ct to pol cooints ov th givn gion G clcult f (,, V G fin th volum of th soli boun b th plns ug tipl intgl fin th volum of th soli boun b th givn sufcs ug tipl intgls in clinicl cooint fin th volum of th soli boun b th plns givn ug tipl intgls in sphicl cooint fin th mss of th lmin gion fin th mss n th cnt of mss of lmin gion boun b th givn gph n its nsit fin th cntoi fo th givn gion
Qustion Evlut th oubl intgl A ov th ctngul gion, :,. A u u u. Substitution Mtho: u u u Qustion Fin th volum of th soli boun b th pln,, n cooint pln. Solution Figu. ( Figu. (b
Volum of soli is givn b, V f (, A wh f (, : Thfo, V f (, A unit.
7 Qustion Evlut u u u Substitution Mtho: u u u
Qustion Lt b gion in th -pln n boun b,, Solution. Evlut A. Figu. A Qustion B ug th oubl intgl, fin th of th gion boun b th cuvs blow: 9,, Solution 9 Figu.
9 A, A A 9 9 9 7 9 unit Qustion B ug th oubl intgl, fin th volum of th soli boun b ths gphs:,,, Solution Figu. ( Figu. (b
Volum, V f (, A A u u - simplif u u u - substitu u ( unit Substitution Mtho: u u u
Qustion 7 Sktch th gion boun b th gphs blow n fin th of th gion b ug oubl intgl in pol cooints :,,, Solution : A, A A = unit Qustion B chnging th intgn fom Ct to pol cooints, vlut : ( Solution :. - Substitution Mtho; u, u u u u
= (b Solution : Whn Whn ln ln ln tn ln sc tn ln sc tn ln sc sc
Qustion 9 Evlut th itt intgl blow : Solution : =
Qustion Clcult G V givn tht th gion G is, G, :,, : Solution : = Qustion Ug tipl intgl, fin th volum of th soli boun b th plns givn blow : ( Clin n plns =, + =. Ellips t th -pln, -intcpt t n -intcpt t. -limit : = - = -limit : -limit : = =. V= V G
- - B A A= with ius cicl of A O ug th substitution of A=
u B = Substitution mtho: u u u u Thfo, V V A B unit G (b =, =, =, + + =. Solution : Whn =, =.,, Whn =, = Whn =, =.,,.,, V V G + =
7 unit (c,, Solution : G V V
( ( B A A= with ius cicl of A B = Substitution mtho: u u u u u
9 Thfo, unit B A V V G (,,, Solution : G V V unit - + =
A of cicl with ius A= B = u Substitution mtho: u u u u Thfo, V V A B unit G Qustion Evlut th itt intgl blow: (
(b ( (c ( ( (
Qustion Evlut th itt intgl blow: ( ( ( ( (b 7 ( ( ( 7 7
Qustion Clcult V G givn tht th gion G is fin s:,, :,, G (
Qustion Ug tipl intgl, fin th volum of th soli boun b th plns givn blow: (,,, Volum, V G A V unit ( ( ( ( ( ( (b Clin + = n plns =, + = =
Volum, V G A V ( ( ( ( ( ( Hlf of of cicl (smicicl with ius i.. Lt wh - - => cicl with ius
Thus, of th sh gion (i. smicicl is unit Altntiv mtho: Fo ( ( - - =
7 Thus, volum, unit V
Qustion Evlut th clinicl cooint intgls blow: ( 9 (b Fo A: Us substitution u u u u u A u u B u
9 Fo B: Thus, = A B Qustion 7 Fin th volum of th soli boun b th sufcs blow ug tipl intgls in clinicl cooint. (,, Volum, V - -
unit (b,, Volum, V ( ( - -
unit Qustion Evlut th sphicl cooint intgls blow: ( sc sc sc sc tn sc tn sc tn A B Fo A: Fo B: Substitut u tn u sc u sc u tn sc tn uu
tn Thus, sc = A B Qustion 9 Fin th volum of th soli boun b th plns givn blow: ( Sph =, con = /, n con = / Volum, V
Qustion Fin th mss of th lmin gion boun b th gph,,, n th nsit (,. = = Mss of th lmin gion (, A 9 9 9 Qustion Fin th mss n th cnt of mss of lmin gion boun b th givn gph n nsit: (,, -is; (,
Mss of th lmin, m ( Cnt of mss = m M m M,, A M, ( 9 (
M (, A 7 7 9 7 M ; m M m 9 9 9 7 Thus, cnt of th mss is 9, 7. (b, cooints; (, Mss of th lmin, m = A Us oubl intgl in pol cooints; Thus, m
( A M Us substitution u u ( u u u
7 A M Us substitution u u ( u u u Thus, cnt of mss = m M m M,,,,, Qustion Fin th cntoi fo th givn gions: ( Th tingl gion nclos b =, = n is.
Cntoi, A A A of, A of, A of = (( A A Thus, th cntoi is,,, =
9 (b Th gion boun b = n = A of gion 9 ( ( A A 9 ( - ( ( ( = (; ( ( = -, =
A 9 ( ( ( Thus, th cntoi is, 9 9, 9 9 9 9, 9 9,