Homework 6 Solutions
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Homework 6, due 12/1 Section 14.4: 2,4,6,8,14,22 Section 14.5: 3,6,14,26,28,36 Section 14.6: 2,7,17,20,38,40 Section 14.7: 5,10,14,16,18,25 Section 14.8: 3,7,15,20,38,39 Section 15.1: 19,20,25,44,59,62 Part I Section 14.4 2) = !"# ! 4) = !"# ! 6) (a) = ! ! 4 ! = !! ! ! ! ! ! = !! ! ! ! ! , = !! ! , !! ! (b) = !"# ! ( ! + ! ) ! = !"! ! !" (2 ! + 3 ! ) ! = !! ! ! !" (3 ! + 2 ! ) , = ! !! ! !!! ! ! ! ! !! ! , ! !! ! !!! ! ! ! ! !! ! 8) (a) = !! ! ! ! = !! ! ! ! = !! ! ! , = ! ! , ! !
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Transcript of Homework 6 Solutions
Homework 6, due 12/1 Section 14.4: 2,4,6,8,14,22 Section 14.5: 3,6,14,26,28,36 Section 14.6: 2,7,17,20,38,40 Section 14.7: 5,10,14,16,18,25 Section 14.8: 3,7,15,20,38,39 Section 15.1: 19,20,25,44,59,62 Part I Section 14.4 2) 𝑚 = !"#
!
4) 𝑚 = !"#
!
6) (a)
𝑚 =𝑘𝑎!𝑏!
4
𝑀! =!!!!!
!
𝑀! =!!!!!
!
𝑥,𝑦 = !!!, !!!
(b) 𝑚 = !"#
!(𝑎! + 𝑏!)
𝑀! =!"!!
!"(2𝑎! + 3𝑏!)
𝑀! =!!!!
!"(3𝑎! + 2𝑏!)
𝑥,𝑦 = ! !!!!!!!
! !!!!!, ! !!!!!!!
! !!!!!
8) (a) 𝑚 = !!!
!
𝑀! =!!!
!
𝑀! =!!!
!
𝑥,𝑦 = !!, !!
(b) 𝑚 = !!!
!"
𝑀! =!!!
!"
𝑀! =!!!"#
𝑘𝑎!
𝑥,𝑦 = !!!!", !!
14) 𝑚 = !"
!
𝑥,𝑦 = 0, !!!!!
22) 𝑚 = !!!!
!
𝑥,𝑦 =8𝑎5𝜋 ,
8𝑎5𝜋
Part II Section 14.5 3) 𝑓! = 𝑓! = 2 𝑆 = 12𝜋 6) 𝑓! = 0, 𝑓! = 2𝑦 𝑆 = !
!6 37+ ln 6+ 37
14) 𝑆 = 2𝜋𝑎! 26) 𝑓! = 2𝑥 − 3𝑦, 𝑓! = −3𝑥 − 2𝑦 𝑆 = 1+ 13(𝑥! + 𝑦!)!
!!! 𝑑𝑦 𝑑𝑥
28) 𝑓! = −2𝑥 𝑠𝑖𝑛 𝑥! + 𝑦! , 𝑓! = −2𝑦 sin 𝑥! + 𝑦!
𝑆 = 1+ 4 𝑥! + 𝑦! sin!(𝑥! + 𝑦!)𝑑𝑦 𝑑𝑥!!!!
!
! !!!!
!
!!
! !!
36) 𝑓 𝑥,𝑦 = 𝑘 𝑥! + 𝑦!
1+ 𝑓! ! + 𝑓!! = 1+ !!!!
!!!!!+ !!!!
!!!!!= 𝑘! + 1
𝑆 = 1+ 𝑓! ! + 𝑓!!
! 𝑑𝐴 = 𝑘! + 1 ! 𝑑𝐴 = 𝑘! + 1 𝑑𝐴 = 𝐴 𝑘! + 1 =
!
𝜋𝑟! 𝑘! + 1 Part III Section 14.6 2) !
!"
7) − !"
!
17) 𝑉 = 𝑑𝑧 𝑑𝑥 𝑑𝑦!
!!!!!
!!!! = !"#
!"
20) 𝑉 = 𝑑𝑧 𝑑𝑦 𝑑𝑥!"!!!!!!
!!"!!!
!!! = 648𝜋
38) 𝑚 = 𝑘 𝑦 𝑑𝑧 𝑑𝑦 𝑑𝑥!! !"!!!!!"!
!!!!
!! = !"#
!𝑘
𝑀!" = 𝑘 𝑦! 𝑑𝑧 𝑑𝑦 𝑑𝑥!! !"!!!!!"
!
!!!
!
!
!=1254 𝑘
𝑦 = !!"!= 2
40) 𝑚 = 𝑘 𝑑𝑧 𝑑𝑥 𝑑𝑦! !!!!!!!
!! !!!!!
!! = !"#$
!
𝑀!" = 𝑘 𝑦 𝑑𝑧 𝑑𝑥 𝑑𝑦! !!!!!
!!
!
! !!!!
!
!
!=𝑘𝑎𝑏!𝑐24
𝑦 = !!"!= !
!
Part IV Section 14.7 5) !
!
10) !"!
!
14) cylindrical: 𝑟!𝑑𝑧 𝑑𝑟 𝑑𝜃!"!!!
!!!
!!! = !!!
!− 2𝜋 3
spherical: 𝜌! sin! 𝜙 𝑑𝜌 𝑑𝜙 𝑑𝜃!!
!!!
!!! + 𝜌! sin! 𝜙 𝑑𝜌 𝑑𝜙 𝑑𝜃! !"!!
!
!!!!
!!! =
!!!
!− 2𝜋 3
16) cylindrical: 𝑟! + 𝑧! 𝑟 𝑑𝑧 𝑑𝑟 𝑑𝜃!!!!
!!!
!!! = !"!
!
spherical: 𝜌! sin! 𝜙 𝑑𝜌 𝑑𝜙 𝑑𝜃!!
!!!
!!! = !"!
!
18) 𝑉 = !!𝜋 4 ! + 4 𝑟 𝑑𝑧 𝑑𝑟 𝑑𝜃!
!! !!
!!! + 𝑟 𝑑𝑧 𝑑𝑟 𝑑𝜃!"!!!
!!! !
!!!
= 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑙𝑜𝑤𝑒𝑟 ℎ𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒 + 4 𝑉𝑜𝑙𝑢𝑚𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑜𝑐𝑡𝑎𝑛𝑡 = !"!
!(2+ 2)
25) 𝑧 = !
!!𝑟! − 𝑟
𝑉 = 4 𝑟 𝑑𝑧 𝑑𝑟 𝑑𝜃! !!!!
!!!
!!!
!!! = !
!𝜋𝑟!!ℎ
Part V Section 14.8 3) 1+ 2𝑣
7) – 𝑒!! 15) 𝐽 = − !
!
4 𝑥! + 𝑦! 𝑑𝐴 ! = 𝑢! + 𝑣! 𝑑𝑣 𝑑𝑢!
!!!!! = !
!
20) 𝐽 = − !
!
𝑦 sin 𝑥𝑦 𝑑𝐴 ! = 𝑣(sin𝑢) !
! 𝑑𝑣 𝑑𝑢!
!!! = 3 (cos 1− cos 4) ≈ 3.5818
38) 𝐽 = 4𝑣 39) 𝐽 = −𝜌! sin𝜙 Part VI Section 15.1 19) 𝑭 𝑥,𝑦, 𝑧 = 6𝑦𝑧𝐢+ 6𝑥𝑧𝐣+ 6𝑥𝑦𝐤 20) 𝑭 𝑥,𝑦, 𝑧 = !
!!!!!!!!!𝑥𝐢+ 4𝑦𝐣+ 𝑧𝐤
25) 𝑀 = 𝑥𝑦!𝑎𝑛𝑑 𝑁 = 𝑥!𝑦 ℎ𝑎𝑣𝑒 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑓𝑖𝑟𝑠𝑡 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑠 !!
!!= 2𝑥𝑦 = !!
!! ⇒ 𝐅 𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑣𝑒
44) 𝐜𝐮𝐫𝐥 𝐅 2,−1,3 = 7𝐢+ 4𝐣− 6𝐤 59) 𝑑𝑖𝑣 𝑭 𝑥,𝑦, 𝑧 = cos 𝑥 − sin𝑦 + 2𝑧 62) 𝑑𝑖𝑣 𝑭 2,−1,3 = 11
