integral lipat dua kuliah k 2 n 3
TRANSCRIPT
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MATA KULIAH MATEMATIKA III( 3 SKS )SEM. GANJIL 2013/2014
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INTEGRAL LIPAT DUA(LANJUTAN)
18092013
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INTEGRAL LIPAT
Integral Berulang
Kita a!at "#$%i$t#&!'ta'ia$ i$t#%&a i!at
*a '#+a%ai ,-*"# a&i +#$a !aat i+aa
!#&"*aa$
(56 ). ( )
∫∫ =
R
dA y x f V ,
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( ) ( )∫∫ ∫ ∫ == R
d
c
b
a
dxdy y x f dA y x f V ,,
( ) ( )∫∫ ∫ ∫ == R
b
a
d
c
dydx y x f dA y x f V ,,
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7-$t-
Hit*$%a
1. a.
P#$6
a.
+. U+a *&*ta$ i$t#%&a$6a
( )∫ ∫ +3
0
2
1
32 dxdy y x
[ ] ( )∫ ∫ +=+3
0
3
0
2
1
2 333 dy ydy yx x
2/45
2
33
3
0
2 =
+ y y
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Ha'i 6a$% 'a"a a!a+ia ita t*a&a$ *&*ta$i$t#%&a $6a
2. Hit*$%a
( )∫ ∫ ∫ +=+2
1
3
0
2
1
3
0
2
23232 dx y xydydx y x
∫
+=
+2
1
2
1
2
2
27
2
6
2
276 x xdx x
2/45=
2 3
0 0
(9 ) y dxdy−∫ ∫
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S-a2
3
4. T#$t*a$ ,-*"# '*at* +#$a !aat6a$%
t#&#ta i+aa !#&"*aa$
a$ iata' !#&'#%i !a$:a$%
24 z x y= − −
{( , ) 0 1, 0 2} R x y x y= ≤ ≤ ≤ ≤
4 2
2
2 1
( 3 ) xy y dydx+∫ ∫
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;#$t* %&a
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G&a
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Maa *$t* i"!*$a$ '##&a$a
U$t* i"!*$a$ '##&a$a 6 aaa
( ) ( )∫∫ ∫ ∫ == sd
c
dxdy y x f dA y x f V
2
1
,,
ψ
ψ
( ) ( )∫∫ ∫ ∫ == s
b
a
dydx y x f dA y x f V 2
1
,,
φ
φ
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7-$t- '-a
=. Hit*$%a i$t#%&a +#&*a$%
P#$6
∫ ∫ 1
0 0
2
2
y
xdxdy ye
[ ]∫ ∫ ∫ =1
0
0
1
0 0
2
2
22 dy yedxdy ye y x
y
x
( )∫ ∫ ∫ −=−=1
0
1
0
1
0
0 22222
ydydy yedyee y y y
[ ] [ ]∫ ∫ −=−=1
0
1
0
1
0
21
02 ye ydydue uu
211 −=−−= ee
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Latia$(P.R)
>.
?.
8.
32
1
( )
y
y
x y dxdy+∫ ∫ 2
2
2
0 2
x x
x
xdydx+
∫ ∫
1 32
0 0
x
x ydydx∫ ∫
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9. G*$aa$ i$t#%&a i!at *a *$t* "#$#t*a$,-*"# a&i t#t&a#&-$ 6a$% i+ata'i -#
+ia$%+ia$% --&i$at a$ +ia$%P#$6
P#&!-t-$%a$ '*"+* 4
P#&!-t-$%a$ '*"+* 6 6 2
P#&!-t-$%a$ '*"+* @ @3
Da#&a '#%iti%a +ia$% 6"#"+#$t* aa' t#t&a#&-$ i
a"+a$%a$ #$%a$ S. Kita aa$"#$a&i ,-*"# i+aa !#&"*
aa$
012463 =−++ z y x
3
2
4
S
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Da&i !#&'
a$ iata' a#&a S
M#"-t-$% +ia$% 6 !aa
S a!at i!a$a$% '#+a%ai
Hi"!*$a$ '##&a$a
Hi"!*$a$ '##&a$a 6
012463 =−++ z y x
y x z 63124 −−=
( ) y x z 244
3 −−=
2231261263
x y x y y x −=⇒−=⇒=+
y x y x 246123 −=⇒−=⇒
( ) }{ 20;240;, ≤≤−≤≤= y y x y xS
{( , ) 0 4, 0 2 }2
xS x y x y= ≤ ≤ ≤ ≤ −
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Jai -*"# a&i +#$a !aat aaa
( )∫ ∫
−
−−=
4
0
22
0244
3
x
dydx y xV
[ ] ( )∫ ∫ +−=−−= −
4
0
4
0
222
0
2 81616
34
4
3dx x xdx y xy y
x
4
0
32
3
1416
16
3
+−= x x x
43
4
4416
3 333
=
+−=V
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Latia$ '-a
Ga"+a& B t#$t*a$ 5 :ia
10. R a#&a 6% i+ata'i-# 05
C5 6 0 a$ 6 'i$
.
11.
12. R '#%iti%a #$%a$ titi
( , ) R
f x y dA∫∫
( , ) f x y y=
2( , ) f x y xy=
( , ) 4 f x y x y= − 2{( , ) 2 , 0 2} R x y y x y y= ≤ ≤ ≤ ≤
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INTEGRAL LIPAT DUADALAM KORDINATPOLAR/KUTUB
23092013
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I$t#%&a Li!at D*aaa" -&i$at !-a&
(&5 θ) !a'a$%a$ -&i$at*t*+/!-a& a&i P
P(&5θ )
&
θ
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Li$%a&a$ +#&!*'at i (050)
• K-&i$at 7a&t#'ia$ F 6F aF
• K-&i$at P-a&/ *t*+ & a
a
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Integral Lipat Dua dalam Koordinat Kutub
K*&,a*&,a t#&t#$t* !aa '*at* +ia$% '#!#&ti
i$%a&a$5 a&i-i5 a$ "aa& #+i "*aiit*$% #$%a$ "#$%%*$aa$ --&i$at *t*+.
Maa ,-*"# +#$a !aat i
+aa !#&"*aa$ i$i a$ i ata' R
i$6ataa$
Daa" --&i$at *t*+5 !#&'#%i !a$:a$% *t*+ R
( )∫∫ = R
dA y x f V ,
( ) }{ β θ α θ ≤≤≤≤= ;;, br ar R
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i"a$a a ≥ 0 a$ 2
Maa ,-*"# aa" --&i$at *t*+
SAL
( ) ( )θ θ sin,cos, r r f y x f z ==
( )θ ,r f =
( ) ( )∫∫ ∫∫ == R R
rdrd r r f dA y x f V θ θ θ sin,cos,
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7-$t- '-a
T#$t*a$ ,-*"# a&i +#$a !aat iata'
!#&'#%i!a$:a$% *t*+
a$ i+aa !#&"*aa$
P#$6
Di "aa "aa
( ) }{ 4/0;31:, π θ θ ≤≤≤≤= r r R22 y xe z +=
222 r y x =+ 2r e z =
=V ∫ ∫ ∫∫ == +
4/
0
3
1
222π
θ rdrd edAeV r
R
y x
[ ]∫ ∫ ∫ ==4/
0
3
1
4/
0
3
12
1
2
1π π
θ θ d edud e uu
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a$:*ta$
( ) ( )[ ]∫ −=−=4/
0
4/
0
1919
2
1
2
1π
π θ θ eed ee
( )19
8ee −= π
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I$t#%&a K*t*+ Hi"!*$a$ U"*" SU$t* i$t#%&a *t*+ ita #$a i"!*$a$'##&a$a r a$ i"!*$a$ '##&a$a O .
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Maa
7-$t- '-a
Hit*$%a i"a$a S aaa a#&a i *a&a$
!#&ta"a 6a$% +#&aa i *a& i$%a&a$ r = 2 '#&ta i
aa" a&i-i
P#$6##'aia$
( ) ( ) ( ) }{ β θ α θ φ θ φ θ ≤≤≤≤= ;:, 21 r r S
( ) ( ) ( )}{ r r br ar S 21;:, ψ θ ψ θ ≤≤≤≤=
∫∫ R
ydA
( )θ cos12 +=r
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;#&a'a&a$ %a"+a& i +aa i$i "aa
S aaa i"!*$a$ '##&a$a r
θ sinr y =2/0 π θ →=
( )θ cos122 +=→= r r ( )
∫∫ ∫ ∫ +
=S
rdrd r ydA
2/
0
cos12
2
sin
π θ
θ θ
( )
∫
+
=
2/
0
cos12
2
3
sin3
π θ
θ θ d r
( )( )∫
−+=
2/
0
33
sin3
2
3
sincos12
π
θ θ θ
θ d
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( )
−+=
∫ ∫
2/
0
2/
0
3sinsincos1
3
8 π π
θ θ θ θ θ d d
−−= ∫ ∫
2/
0
2/
0
3 sin3
8 π π
θ θ d duu
( )2/
0
4coscos1
4
1
3
8 π
θ θ
++−=
( )( ) ( )3
22121
4
1
3
8 44 =
−+−−=