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Scanned by CamScanner L i J* iL Z Cc u J o I JA A 5 2 5 T U n i v e r s i t la d c Fcder n 1 F 1u 1n i n c n s c E G HI l ï t i t ut o t le is l a t e r , r ai t Ic n e E s t a t fs Ai c n L S y 2 0 1 5 2 G M A D o p n r \ n n i c n t o r lp Mn ï e mt i c a A p l i c a d a I nt cgml Ti pl a z O C M c u l e v 1 v o n d e t V a r c g i ño d o p r i n i c i r o o c l n Ht e , hm i h r d a p o r r · 1 U , u z , O e C a l c u l e Jn , m , o n de w . . s óhd o de h n Ä a d o p o r . o , y o , Æ v 2 , z Y v 2 e z 0 4 C a l c u ta JJH , y F M , o n d e w n r c g iño dch m i t ñó p o r r o , z o , r z V c z 5 C a l c u le Jí f , z d v , o n d e W e s t D s i t u a t l o n o p r i m c i r n o c Aa n t e , l i m i t a d o p e l o s p l a n p s z - O, £ - O, y 2 Æ e p e l o c M n d m y 2 2 4 ü M o u l e JM . . z d v , o n d e V . . Só hd o h i n h a d o p e l o s p h n o s c O, y O, ' O, V z 1 c 7 C a l c u l e Jf f , 2 4 z [ l l ï , o n d e ì ï ï ei o s ó id ü l i m i t a d o p o r = + j / + z 2 , x - n , V - O, z = O, e z - 1 8 U s e u m i A i n t e g r a l t r i p l a p a r a c a lc u l a r o v o l u m e d o só l id o W l i m i t a d o p o r W z 4 x 2 y 2 e z y , s h u a d o n o i n t e D o r d u c M n d r o 3 2 v 2 1 z 0 (c ) z - 3z 2 , z - 4 x 2 U n c z + V - 0 W y O z O c y 2 , 2 y % v z z 2 4 , n o p r i n l e ó o c Aa n m 9 C a l c u l e Jf í . z d v , o n d e W o s ól i o l i m i t a d o p e la s s u p e r f íc ie s z - . , z (z + y 2 ) c z 2 + y 2 + z 2 _ 4 1 1 C a l c u l e j I f . J · + 3/ 2 { l v , o n d e ì v , ) s ól id o l i m i t a d o p e la s s u p e r f íc ie s z = ( Z y 2 ) , ï 2 . ° C o n s i d e r e a n Ae gFa l \ t e > a t \ n 0 - I J , - - , - , t zdYd= E x p r e s se I em c o o r de n a d a s c Mn ó c a s e c a k u l e , s c u v n m . (b ) E x p r e s s e I c l n c o o r d e n a d a s c & f r i c a s c c a l c u l e o s c u v a lo r 13 C a lc u l e o v o l u m e d o sá . id o W q u e e s t á de n t r n r la e s fc r n 2 + t12 + z 2 _ 4 , a c i m a d o p l a n o 2 - 0 c a m x o d o c o n e z c m 14 C a l c u le o v o l u m e d o s d l id o W ( la do p o r x 2 + ?/ 2 + (z 1)2 [ 1 c z > c2 + y 2 15 C a l c u la a n i a s s a d o s Ä d o W m u n do n u p r hn d r u u c t a n t c , h m k a d u p e lo s p L1n o s æ O, v 2 z c p c l u c i l h n h u y 2 z 2 2 , s u p u u d u q u u i L d c u s id aBde n u p m u þ , y , Z D Wupur =i ün1l % d is M n c hB d e s t e 1J u 1t l u n o p l a n o a ?j Scanned by CamScanner C á1c u 1o I I I A 2 0 15 2 6 l c u l c a m i w w d o s ó l id o ï V , 0i m ï t a d u p e l a s _ u p c r Hc 0c s x 2 + y 2 _ 1 , z + 3/ - 2 c z + (' , Sc a d c n s i d a d o c m (z , y , z ) é d a d a p o r p ( c , y , z ) = z ° C a l m e a I n a s s a d n sóh d o h n Ä a d o p e lo p l a n o o , o c M n d m z 2 y 2 2 z e peo q n e z J · y z , s c a d e n s id a d e p (Æ , y , z ) = r 2 + t/ 18 C a l c u le a m a s s BL d o s ó l id o l i m i t a d o s u p e r in n n c n l c p o r n 2 + v 2 4 ( · 1 ) 2 _ 1 , i n f e r i o r m e n Ae p o r z = (1 2 + :rl | ) , s e n d o a d e n s id a d e ig u a l a o q u a d r a d o ,-l n d is A n c i n d c (z , y , z ) a o l l a n o z - n 1 9 C a l c u l e n I l l a s s a d o s ól id o ïV : = 2 + 4/2 [ 1 , = 2 + V + z 2 < 4 c 0 , s e n d o a d n s id a d e d a d a p o r p (Æ , y , z ) = 2 2 o E n c o n t r e a n m s s a d a r c g ñ o s óh d a h n Ä a d a peas s n p c r f íc k s z 16 2Æ 2 2 y 2 c 4 2 x z 2y 2 s e a d e n s id a d e d n SÆ d o é ó , y 2 1 E n c Ä , r e 0 mo n 1 M 0 d c i n r c h . Do s áhd o n o p r h n d r o oc a n t e , ' hn Mad o p e l a s u p c r f : c i e s Z y , n 2 y 2 , z O c x O , s e n d o n d c n %hM c d a ó p o r p cr , u , 4 k z , o n d e k C u m n c o n M a ó 22 C a o n Ä u ì d o c e n t >u dc m u s s a d o Sá Hd u W d a d o p u r u2 l y 2 z 2 | 2 a , z 2 y 2 I z 2 Z 8 z , z = - z [ l 3 ( · z + y z ) , s c a d e n s id a d e 11o p u n t o (z , u : z ) i n v e r s a m e n t e p r o p o r c i o n a l a o q u a d r a d o d a d is t i i n c i a d o p o n t o $t o r ig e m 2 3 C D u s IDc r c o c i l i n d r o h o m o g ên e o = 2 + (u n ) 2 [ a c o [ z < I I C a l c u l e o i nomnt o d e i n r c i a e l n 1' " l açi o a o e ix o z , c m f u n ç o d a m a s s a M d o c i l i n d r o 2 1 s c ï n I = = J . J , ' d z ây ó R = c s c r c v B\ ah\ \ =gT . . " d . . ' " . 2 5 Se j a ( , v o l u m e 9l o s d l i d o W c o i n t i m S f c > n s x 2 + J12 + 2 _ 1 c x 2 + V 2 + z 2 - 2z E x p r e s s e (5 c l n Cal cuar o v d u m e ó w c m (a ) e l n c o o r d e n a d a s c i l ín d r i c a s n a o r d e r u d z d r r to (b) e m c o o r d e n a d a s e s l r i c a s n a o r d e m t 1pddo Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner _ · - - · Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner ' ' · i · Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner
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