Final Practice Math 2400-004, Calculus III, Fall 2019
Name:
1. Let ~F(x, y, z) = hx, y, xezi and let S be the part of the cylinder x2+y2 = 1 beneaththe plane z = 2 and in the first octant, with positive orientation. CalculateZZ
S
~F · d~S.
sfIM%9IyTho ) -
- Mosca ,sie cos
,h >
OE h I 2
OI a E TY 2
The To ,o
,D
÷ e. sina.co . ..
.
" I:.
÷.
:/= f- Cos 6
,
- sin a,
O )
s
I -
- Loso ,since
,
o >Wrong orientation !
2
Fok So f cos a,
since,
cos a e
" " 7 . faso ,since
,
o ) ah do
2
= S cos'
a t sin'
a dh da
O
=
1%2da =
2. Let ~F(x, y, z) = hesin(y), xz + 2, sin(y)x2 + zi and let S be the paraboloid y =
1� x2 � z2 with y � 0, with positive orientation. Calculate
ZZ
S
~F · d~S.-
-
①Almost closed !
§ E. dj = Sffdiver ) -
Sgs,
I . di ②
s'
is
① div (E) = O to t ITho ) stresa ,
o
,rsina >
OE r Elyazzie= I
¥ f! Got
' "
r dydrdd
= zit So
'
r - r
'
dr =2x ( E - ⇒ to
'
= IL
run
② F l riot -- Tresa
,o
,r since >
are
0=-012-4
Is fo,
- r, 07
fo"
fo
"
f . . .
,
rios asin at 2,
. - - ) - To,
-
r,
07 do do
= SoilSo"
- r
'
Cosas in a - 2 r drove
= so"
- confine - I do
go ¥ - f - zH=5= - 2T
3. Let ~F(x, y, z) = hyexy � zy, xexy � xz,�xyi and let C be the intersection of thecylinder z2+y2 = 9 and the paraboloid x = y2+z2, oriented clockwise. CalculateZ
C
~F · d~r.
f- His ,zI=e×t - xyz
Fflxihztefye " 't- yz
,text - xz
,
- X 's >
IF- is conservative & C is a loop,
therefore
si÷
4. Let ~F(x, y, z) = hyexy � yz, xexy � xz, 1 � xyi, and let C be the curve ~r(t) =ht cos(t), t sin(t), ti with 0 t 2⇡. Calculate
RC~F · d~r.
f- ( x, y
,z ) = eat - XYZ t Z I conservative
F ( o ) = fo,
o
te"
- TZ,
teth - xz,
I
_¥F Kal -- Eh ,
o,
2K )
By F- To LI
{ F. di -
- f ka,o
,it - flo
, 0,4
= ( I t 2T ) - I
=
5. Let ~F(x, y, z) = hx cos(x), xy � z, ez + yi and let C be the lines connecting(1, 0, 0), (1, 1, 0), (1, 1, 1), and (1, 0, 1), oriented counterclockwise. CalculateZ
C
~F · d~r.F is not conservative
¥¥,Stokes'Theoren
MagtfThe ) - Tl
,is
,z >
S is green0 EYE I
,o E ZE I
} £
I = ft,
0
,o )
curl
h⇒=µ.I
.In .
xoxo ) as - z ezty) -
- T I - I
,0
,y )
= To,
o,
Y ?
f! fol fo,
O,
y ) . f I
,o
,o ) dy d
2--00
6. Let ~F(x, y, z) = hy+cos(x), cos(y)ey, z2�xi and let S be the surface of the regionbounded by the parametric equations
~r(u, v) = h cos(u)(2 + sin(v)), sin(u)(2 + sin(v)), cos(v)i
with 0 u ⇡2 and 0 v 2⇡ and the plane y = 0, positively oriented.
Calculate
ZZ
S
curl ~F · d~S.
It's a quarter
-of a torus
=i Thin
-
Amateur Boundary is circle of radius
Sketch Not.
✓ f included
I catered at ( o,
2,
o ).
Use Stokes ' Thin to replace S
I I-
with S ' where S 'is given by
INMFfr
,o ) .
- fo,
2 those,
Tina >
OE REI
OE OE LTI
until '
µ.In
.
'
Ts.
Hashi coaster E- x) = TO,
l,
-
I >
To fr, 0,07
) ! go'
fo ,I
,
- I > a Er, 0,0 ) dr de
= @
7. Evaluate
ZZ
S
~F · d~S where ~F(x, y, z) = hey2 + xy, y2 + z, ex2 � zyi and where S
is the positively-oriented boundary of the region bounded by and including thefollowing sides: z = 0, x = 1, y � 0, x = y2, and z � x = 1.
#y
div (E) = Yt 2g - y = 2g
Divvgacethnm!
Jo f! f!"
2g az dx dy
= So
"
Sy! 2g t Zyx dxdg
-
- So
'
2 yx t y x
'
I'
y.
dy
= So
"
( 2g t y ) - ( 2g'
t y' ) D8
= fo'
3g - 2g'
- g5
dy
-
-
Is - I - It !-
- z - E - to =
8. Evaluate
Z
C
~F · d~r where ~F(x, y, z) = h sin(x2), yz � ey, sin(y2) � xi and C is the
boundary (oriented counterclockwise) of the portion of the surface x2 + y2 = 9(including z = 0) cut o↵ at the plane y + z = 4.
ancient;
.: .:* fi
Sir yz - et sidi . a
)
New surface= 12 yoshi ) - y
,
I
,O >
bFfr ,
a) of rosa ,r since
,4- rsina )
OF r E3OEOEZTI
l÷÷÷÷÷÷ .to .ms
So"
f! f . ..
,I
, 07 . to,
r,
r ) ddr da
= zits ! r ar
-
- La C El ! ) -- ⑦
9. Evaluate
ZZ
S
ez dS where S is the positively-oriented surface given by ~r(u, v) =⌦cos(u), sin(u), uv2
↵with 0 u, v ⇡.
Icetar field
wtf matter
Ti f - sink ),
costal,
I 7
I . T 0,
o,
I 7
Irixit-
-
tis ;"÷ in
'
yanquiis ',071
=-
- I
So
"
fo
"
use"% dr duw -
- ¥
dw = Uz d N
got'
f!÷ ew aw da
= got'
e" "
- I du
= I ¥ e
" "- 4) I !
-
- ÷E%-a
10. Evaluate
Z
C
~F · d~r where ~F(x, y) = h cos2(x) + y2x, x2y + x � ey2i and C is the
path starting at (0, 4), going down the y-axis to (0,�4), traveling counterclockwisealong x2 + y2 = 16 to (4, 0), then following y + x = 4 back to (0, 4).
" " "
, ,
Greenish
A- - 4T g Ty= Zyx
O
¥ = Zxytl
SfZyx - ( Lyx t 1) DA
= Sfg- I DA
= -(8t
11. Evaluate
ZZ
S
~F·d~S where ~F(x, y, z) = h�z, ex2, y2i and S is the positively-oriented
surface formed by gluing together the cylinder x2 + y2 = 9 with 0 z 9 withthe elliptic paraboloid 5+x2+y2 = z with 5 z 9 and the surface given by theparametrization ~r(u, v) = hu cos(v), u sin(v), 9i where 0 v 2⇡ and 2 u 3.
L Rough
⑥ sketch did E) = O
\
"
""
!I
S -- bottom circle filled to a disc
Tatton openFlore) -
- Crusoe,
rsino,
O >
T -
- fo,
0
,- r >
.
E- RE 3
OEO -22T ,
S
"
f . . .
,.
..
,
artiste ) . 40,0 ,- r > area
O
= Song! -r' sinha draa
2T
=- Eff
"
sin'
oda
of( l-{) da
O
=- ILE- sinful ) ) !
4
=
- site4
So drifter- If Eas = 8
12. Evaluate
ZZ
S
~F · d~S where ~F(x, y, z) = hxey2 , y sin(z), 2x� zey2i S is the surface
in the first octant formed by the planes y = 0, x = 0, and x+ y + z = 1 orientedoutwards. &
¥sdid El -
- et 't sink ) - es'
= sin Cz ).
to Got" t
sink ) dzdydx
= fo'
fo' - ×
- cos Iz ) to
" " "
dos di
=
So'
So
' - ×
I - cos C I -
x - y ) dis ox
-
- fol y tsin I I - x- 2) I dx
--
So'
I - x t sin ( o ) - sin I I - x ) dx
-
- x - E - costa !!= I - z - cos to) t cost 1)
=cosll
13. Evaluate
Z
C
~F ·d~r where ~F(x, y, z) = hzex, sin(z), ex+y cos(z)i and C is the curve
given by ~r(t) = ht2 cos(t), t, ti where 0 t 2⇡.
ftp.zl-zettysinlz )
Tfhgy,
z ) -
- feet,
sinlz,
extycoslz ) ) IF- is conservative
.
Flo ) -
- To,
o
,o ) F ( 2×1=4442, 292T >
By FTOLI,
{ F. di -
- f Hi,
2x,
2x ) - flop ,o)
= Zte" 't 21T Sir I HI)
= 2te
14. Evaluate
ZZ
S
curl(~F) · d~S where ~F(x, y, z) = hy,�x, zi and S is the surface, ori-
ented outwards, made of the half-cone z =px2 + y2 with 0 z 1 and the
cylinder x2 + y2 = 1 where 1 z 4.
ipscar " " -
-
ppg!;'
if;) .
. to ,o
,-
i - is
= To,
o, -27
Replace S with s'
g Ftr ,a ) -
- Kroos a,
r since,
47
ITiko
,o
,
-
r >
I"
fo
"
To,
o,
-27 . to,
o,
- r > dr da
= 2T So
"
2 r dr
=
15. Evaluate
Z
C
x ds where C is the curve y = x2 in the xy-plane from 0 x 1.
dScalar field !
µHtt - It ,t7
OEEII
FIH .
- fl ,2T >
IF Ith -
- TE
So
"
tint at a- Hat'
du - stat
= IS! n' "du
. int .
16. Evaluate
ZZ
S
curl(~F)·d~S where ~F = h cos(ey2+z2+x2), esin(xy), sin(cos(sin(cos(sin(xyz)))))i
and S is the surface of the sphere x2 + y2 + z2 = 4 above and including the planez =
p3 oriented outward. -
-
This is a closed surface.
div I curl LED = O
By Divergence Theorem
I! curl IE ) . DJ = ②