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a

I

I Ltp x2y3 It Byod jdi docI tdyyT di Lt B oilydoc By

3 25

Along x 5g doc 5dgii dI CLtMC5yYg3Gdy t Py 5g2dg

125 Lt B y dy t 25Bysdy25 52 6B y dy

y o I

Si di fo 25652 6B y dy25 FL t B

org

Q2Stokes Theorem TAT da ft DT

Iit

Convert it to cylindrical

5 25 sino cos05 sindOT

5z Sindsi cos018 Its cosofE

J di C 25sins lots 2 cosd sold f I DZ O S R

Si di R fo 2sin'dold LDx i o 5 I Co ily Co 4g I

5 I J Liye5 costs sin 08 sind5 toosold

4s sin I

die sdsdfE

T x il do 4 s s inddsdd

SC th ut da If for 4 s sind dsdd

Oz

Q3

i

3x't3yCartesian to Cylindrical

x scosol yessing 2 Z

D J _350501 3555435 cosittsintftp.T do fo5fYf23s2 sdsddd2

5x2iTxo3sBds

louf3sI 1207

Now ftp.t dE fosi da

Convert it to cylindricalE cos45 sin ofof

of sings cos OfE I

J s3 cos ft sin4015 t s osdsindfcosbtsinH.JO

Three sides to cylinder8

G Top dei sdsDIET da O

2 Bottom dei sdsdf IJ dot O

3 side do sold diesii dei s cos40 tsin 40 ddd2

Si di fo fo s cos I tsin 4 dddZ

Note 5 2 fo dz 5

Si dei 16 5 f cos oldolt sin oldd

80 77 47 112074

Question 3

147 ut de f i do

it r3indrthr2coscetr2tandFl7

J rIgdCr2 r's in d

rsitnaFee since4 rEosa

rateof Catana

D J Cupsin d trY Goshsince

sinYet coshe since

Coste4 r

since

Question 4

ftp.utdt f urgII r2sinAdrdddd

o7ir3dr Fco5addf a

R se Az singe

LIR E singes

TI Cit t 3FL

I 2tt3r

Surface consists of two parts

G The icecream cone

r Rid O 2E A o o

doe R'sina.dd.deri

da CRsind RsindddddR4sin2ddddO

fu da R f sin'd 1 01

R Czaffaz sina.IQ ob

2itR4CE sinEs

tfICt 3Fsz

do

2 The core

A If I O 72T r o R

dersinddrdddudw 4sinclcosdrsdrdd

e.fr dro

Su doi fsl r3dr o

dot Bzer

Therefore

So.da tRI Iz Bz tfs

t 2rt3fsT

d

Question 3 Archimedes

Lets define the pyramid to have

vertices at

Como Ca a b Ca a b Ea a b Gaia b

Calculate for this surface

The force on the bottom surface is

p egb die dedge

E fpdoi faaf.aegbd.edu I

Question 5

E b l

4a'begE

The force on a tilted surface can be

found by considering the equation ofthe surface surface in diagram

Z hat x O

The normal to this surface is justthe gradient of the scalar function

2 Z C

ri p Cz IaEE EE

Normalising we have of at battab

K

B

dz2 Laxda da doc

ota dydlridyd.IAEbx.azdyftabd.caEbxraztbz

adydx.cat BI

Now in ywe are integrating between

the lines yx and y K

E HedaaL f

I f gbzxtadgd.cat boil

easab C z.cyd.cat boil

es f IcaE bail

2egbagCAI BI

On surfaces and the 2

component will be the same as

but the se or y component will

be different

E 2ebICaEtb5E 2ebICaEtbEE 2egeba Cai 551

Summing the forces on all five

surfaces we have

E 4a'begI 8e I3

Is a begI

s Gal beg Ivolume xegI as expected

In general

F Ssp di

SSSepdt

Pp egE

E SSSegdtIeg VI

Esa'beg I