Final Practice Math 2400-004, Calculus III, Fall 2019

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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