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SHMMUMajhmatik Anlush

(Sunartseic Polln Metablhtn - Dianusmatik Anlush)Lseic tou 8ou fulladou asksewn

Askhsh 1. Na upologsete ton gko kai to kntro mzac tou stereo poufrssetai, ktw ap ton kno = =6 kai nw ap th sfara me kntro toshmeo (0; 0; ) kai aktna .

Lsh. H exswsh thc sfarac me kntro to shmeo (0; 0; ) kai aktna enai = 2 cos kai epomnwc o gkoc tou stereo enai

V =

Z 20

Z =60

Z 2 cos0

2 cosd d d =7

123 kub. mon.

Jewrome thn puknthta mzac stajer. Lgw summetrac to kntro mzacja brsketai pnw ston xona z, opte x = y = 0. Gia thn trth suntetagmnhqoume

z =1

m

Z 20

Z =60

Z 2 cos0

cos 2 cosd d d =37

28 :

Sunepc to kntro mzac tou stereo enai to (0; 0; 3728).

Askhsh 2. 'Estw D o monadiaoc dskoc kntrou (0; 0). Prosdiorste toucpragmatikoc arijmoc , gia touc opoouc sugklnei to genikeumno oloklrwmaZ Z

D

1

(x2 + y2)dA :

Lsh. Qrhsimopoiome polikc suntatagmnecZ ZD

1

(x2 + y2)dA = lim

"!0

Z 20

Z 1"

r2(1)dr d

= 2 lim"!0

(1 "2(1)) :

Epomnwc to genikeumno oloklrwma sugklnei gia < 1.

Askhsh 3. Na brete to kntro mzac enc kalwdou puknthtac (x; y; z) =kz kai elikoeidoc sqmatoc pou perigrfetai ap tic parametrikc exisseic

x = 3 cos t ; y = 3 sin t ; z = 4t ; 0 t :

Lsh. H mza tou kalwdou enai

m =

Zr

(x; y; z) ds =

Z 0

(r(t))jjr0(t)jj dt = 10k2 :

1

Oi suntetagmnec tou kntrou mzac enai

x =1

m

Zr

x (x; y; z) ds =1

10k2

Z 0

(3 cos t)(4kt)5 dt = 122

y =1

m

Zr

y (x; y; z) ds =1

10k2

Z 0

(3 sin t)(4kt)5 dt =6

z =1

m

Zr

z (x; y; z) ds =1

10k2

Z 0

(4t)(4kt)5 dt =8

3 :

Askhsh 4. Na upologsete to epikamplio oloklrwmaZr

x2dx+ xydy

pou h kamplh r apoteletai ap to txo thc parabolc y = x2 ap to shmeo(0; 0) wc to shmeo (1; 1) kai to eujgrammo tmma ap to (1; 1) wc to (0; 0).

Lsh. Orzoume

r1(t) = (t; t2) t 2 [0; 1] kai r2(t) = (t; t) t 2 [0; 1]

kai qoume tiZr

x2dx+ xydy =

Zr1

x2dx+ xydy Zr2

x2dx+ xydy

=1

15:

Askhsh 5. 'Estw h sunrthsh

f(x; y) = arctany

x;

h opoa, gia x > 0, enai sh me th gwna twn polikn suntetagmnwn toushmeou (x; y).

(i) Dexte ti h f enai sunrthsh dunamiko gia to dianusmatik pedo

F (x; y) = (y

x2 + y2;

x

x2 + y2) :

(ii) Upojtoume ti A(x1; y1) = (r1; 1), B(x2; y2) = (r2; 2) enai do shmeatou prtou tetarthmorou (x > 0) kai r ma C1 kamplh me arq to A kaiprac to B. Dexte ti Z

r

F ds = 2 1 :

2

(iii) An r1 kai r2 enai parametrikopoiseic tou nw kai tou ktw hmikukloutou monadiaou kklou antstoiqa, me arq to (1; 0) kai prac to (1; 0) nadexete ti Z

r1

F ds = ; enZr2

F ds = :

'Erqetai to parapnw se antjesh me to jemelidec jerhma tou apeirosti-ko logismo gia dianusmatik peda klsewn?

Lsh.

(i) Paragwgzontac qoume ti

rf(x; y) = ( yx2 + y2

;x

x2 + y2) :

(ii) Smfwna me to jemelidec jerhma tou apeirostiko logismo gia dianu-smatik peda klsewn qoume tiZ

r

F ds =Zr

rf(x; y) ds = f(tan 2) f(tan 1) = 2 1 :

(iii) PrgmatiZr1

F ds =Z 0

dt = kaiZr2

F ds = Z 2

dt = :

To parapnw den rqetai se antjesh me to jemelidec jerhma tou apei-rostiko logismo gia dianusmatik peda klsewn afo h sunrthsh f denorzetai gia x = 0.

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