ÿþZ a d . 1
( x 2 + y 2 ) d l A ( 1 , 1 ) ; B ( 4 , 4 )
+"
L
1 = a + b a = 1
ñø ñø
òø4 = 4 a + b Ò! òøb = 0 Ò! y = x
óø óø
y ' = 1
4 4
1 2 6 2
( x 2 + x 2 ) 1 + 1 d x = 2 2 x 2 d x =
+" +"
3
1 1
Z a d . 2
+"( 2 x - y ) d l A ( 2 , 2 ) ; B ( - 2 , 4 )
L
ñøa = - 1
2 = 2 a + b
ñø 1
ôø
2
Ò! y = - x + 3
òø4 = - 2 a + b Ò! òø
2
óø ôø
óøb = 3
1
y ' = -
2
2
2
2 2
1 1 5 5 5 5
ëø ëø öø ëø ëø
÷ø ÷ø ÷ø
+"ìø2 x + 2 x - 3 öø 1 + ìø 2 ÷ø d x = 2 +"ìø 2 x - 3 öød x = 2 ìø 4 x 2 - 3 x öø = - 6 5
íø øø íø øø íø øø íø øø
- 2 - 2 - 2
Z a d . 3
+"( y ) d l y 2 = 4 x O ( 0 , 0 ) ; P ( 1 , 2 )
L
2
ëø 1 öø 1
2
( y ' ) = ìø- ÷ø
=
x
x
íø øø
1 1 2
2
1 + x = t 3
1 + x 4 4
2
t d x = t = ( 8 - 1 )
+"2 x d x = +"2 1 + x d x = d x = d t = 2 +"
x 3 3
1
0 0 1
Z a d . 4
A ( 1 , 1 ) ; B ( 2 , 8 )
+"( y ) d l y = x 3
L
2
2
( y ' ) = ( 3 x 2 ) = 9 x 4
1 + 9 x 4 = t 2
2 1 4 5
1 4 5
1 1 1
2
x 3 1 + 9 x 4 d x = = d t = t 3 = ( 1 4 5 3 - 1 0 3 )
1
+" +"t
1 0
1 8 5 4 5 4
x 3 d x = t d t
1 1 0
1 8
G r z e g o r z M r z y g Bo c k i , W I L i Z, s e m . I I I , g r . 2 1
Z a d . 5
ëø öø
x 1
+"ìø ÷ød l y = 1 d" x d" 2
ìø ÷ø
y 4 x
íø øø
L
2
öø
2 ëø - 1 1
( y ' ) = =
ìø ÷ø
x 2 x 4
íø øø
x 4 + 1 = t 2
2 2 1 7
1 7
x 4 + 1 1 1 1
2
x 5 d x = x 3 x 4 + 1 d x = = d t = t 3 = ( 1 7 1 7 - 2 2 )
1
+" +" +"t
2
x 4 2 6 6
x 3 d x = t d t
1 1 2
2
Z a d . 6
À
+"2 y c o s ( x ) d l y = s i n ( x ) 0 d" x d" 2
L
2
( y ' ) = c o s 2 ( x )
À
2 1
2
1
1 + c o s 2 ( x ) = t 2 2
2
2 = - 2 d t = - t 3 = ( 2 2 - 1 )
+"s i n ( x ) c o s ( x ) 1 + c o s 2 ( x ) d x = +"t
2
c o s ( x ) s i n ( x ) d x = - t d t 3 3
0 2
Z a d . 7
9
y = x x 0 d" x d" 4
1 + x d l
+"
4
L
2
3 9
2 ëø öø
( y ' ) = x = x
ìø ÷ø
2 4
íø øø
4
4 4
ëø öøëø öø
9 9 9 9
ëø ëø öø
÷ø
+"ìø 1 + 4 x ÷øìø 1 + 4 x ÷ød x = +"ìø1 + 4 x öød x = ìø x + 8 x 2 ÷ø = 4 + 1 8 = 2 2
ìø ÷øìø ÷ø
íø øø íø øø
0 íø øøíø øø 0 0
Z a d . 8
+"( x + y ) d l O ( 0 , 0 ) ; A ( 1 , 0 ) ; B ( 0 , 1 )
L
ñøl : y = - x + 1 0 d" x d" 1
1
ôø
ôø
y = t 0 d" t d" 1
òøl : x = 0
2
ôø
ôøl 3 : y = 0
x = t 0 d" t d" 1
óø
1 1
+"( x - y ) d l = +"( x - x + 1 ) 1 + 1 d x = 2 +"d x = 2
l 1 0 0
1
1
+"( x - y ) d l = +"t 0 + 1 d t = 2
l 2 0
1
1
+"( x - y ) d l = +"t 1 + 0 d t = 2
l 3 0
1 1
+"( x + y ) d l = 2 + 2 + 2 = 1 + 2
L
G r z e g o r z M r z y g Bo c k i , W I L i Z, s e m . I I I , g r . 2 2
Z a d . 9
1 1 À
2
+"6 x y d l x = 3 c o s ( t ) y = 3 s i n ( t ) 0 d" t d" 4
L
2
2 îø- 1 1
( x ' ) = s i n ( t ) ùø = s i n 2 ( t )
ïø úø
3 9
ðø ûø
2
1 1
2 îø
( y ' ) = c o s ( t ) ùø = c o s 2 ( t )
ïø3 úø
9
ðø ûø
À À 2
5
4 4 2
2 2
s i n ( t ) = u
1 1 2 2 2 2 2
2
6 c o s ( t ) 1 s i n 2 ( t ) d t = = u 2 d u = u 3 2 = =
+" +"c o s ( t ) s i n ( t ) d t = +"
0
3 9 9 2 7 c o s ( t ) d t = d u 2 7 8 1 6 4 8 1 6 2
0 0 0
Z a d . 1 1
x z 2
; z
+"1 + 2 y d l x = t ; y = t 2 = 3 t 3 0 d" t d" 1
L
2
( x ' ) = 1
2
( y ' ) = 4 t 2
2
( z ' ) = 4 t 4
1 1 1
2 t 4 2 t 4 2 2 2
4
( 1 +
+"1 + 1 + 4 t 2 + 4 t 4 d t = 3 +"1 + 2 t 2 2 t 2 ) d t = 3 +"t d t = 1 5
3 2 t 2
0 0 0
Z a d . 1 2
; y ; z
+"( x y ) d l x = e t = e - t = 2 t 0 d" t d" 1
L
2
( x ' ) = e 2 t
2
( y ' ) = e - 2 t
2
( z ' ) = 2
2 2
1 1 1 1
( e 2 t ) + 2 e 2 t + 1 ( e 2 t + 1 )
t
e 2 t + 2 + e - 2 t d t = d t = d t =
+"e Å" e - t Å" 2 + e 2 t + e - 2 t d t = +" +" +" 2
e 2 t
( e t )
0 0 0 0
1 1
1
e 2 t + 1
= d t = ( e t + e - t ) d t = ( e t - e - t ) = ( e - 1 - e - 1 + 1 ) = e - e - 1
+" +"
0
e t
0 0
Z a d . 1 3
3
x ( y + z ) d l x = c o s ( t ) y = s i n ( t ) 0 d" t d" 2 À
z = t
+"
4
L
2
( x ' ) = s i n 2 ( t )
2
( y ' ) = c o s 2 ( t )
9
2
( z ' ) =
1 6
G r z e g o r z M r z y g Bo c k i , W I L i Z, s e m . I I I , g r . 2 3
2 À 2 À
öø s i n ( t ) = u
îø 3 ùøëø 9 5 3
ëø öø ëø
=
÷ø
ïø
+"ðøc o s ( t ) ìøs i n ( t ) + 4 t ÷øúøìø 1 + 1 6 ÷ød t = 4 +"ìøc o s ( t ) s i n ( t ) + 4 t c o s ( t ) öød t =
c o s ( t ) d t = d u
íø øø íø øø
ûøìø ÷ø
0 íø øø 0
0 2 À 2 À
p = t q ' = c o s ( t )
îø ùø
5 1 5 1 5 2 À 1 5 2 À
= ( t Å" s i n ( t ) ) -
ïø úø
+"( u ) d u + 1 6 +"t c o s ( t ) d t = 0 +"s i n ( t ) d t = 1 6 c o s ( t ) = 0
0
4 1 6
0 0 ðø 0 ûø
p ' = 1 q = s i n ( t )
Z a d . 1 4
+"( z ) d l x = t c o s ( t ) y = t s i n ( t ) z = t 0 d" t d" 1
L
2 2
( x ' ) = ( c o s ( t ) - t s i n ( t ) ) = c o s 2 ( t ) - 2 t c o s ( t ) s i n ( t ) + t 2 s i n 2 ( t )
2 2
( y ' ) = ( s i n ( t ) + t c o s ( t ) ) = s i n 2 ( t ) + 2 t c o s ( t ) s i n ( t ) + t 2 c o s 2 ( t )
2
( z ' ) = 1
1 1
3
t 2 + 2 = u 2 3
1 1
2
( 3
+"t 1 + t 2 + 1 d t = +"t t 2 + 2 d t = = u d u = +"u d u = 3 u 3 = 3 3 - 2 2 )
2
t d t
0 0 2
Z a d . 1 5
L = y = l n ( x ) 2 d" x d" 5
+"d l
L
1
2
( y ' ) =
x 2
1
5
2
u = ( x 2 + 1 ) v ' = l n ( x )
5 5 5
ëø öø
1 x 2 + 1 d x
ìø ÷ø
L = 1 + d x = = - =
+" +"l n ( x ) x 2 + 1 d x = +"
x 1
ìø ÷ø
x 2 x
x 2 + 1
2 2 2
u ' = v =
íø øø
2
x
x 2 + 1
5
ëø öø
2 6 5 2 6 5 2 6 5 2 + 5
÷ø
= - - l n ( x + x 2 + 1 ) = - - l n ( 5 + 2 6 ) + l n ( 2 + 5 ) = - + l n ìø
ìø ÷ø
5 2 2 5 2 5 2
5 + 2 6
íø øø
Z a d . 1 6
3
L = x = 7 c o s ( t ) y = 7 s i n ( t ) 0 d" t d" À
+"d l
4
L
2
( x ' ) = 4 9 s i n 2 ( t )
2
( y ' ) = 4 9 c o s 2 ( t )
3 À 3 À
4 4
2 1 À
L = 4 9 d t = 7 =
+" +"d t
4
0 0
Z a d . 1 7
r ( ³ ) = 1 + c o s ³ 0 d" ³ d" À
x = ( 1 + c o s ³ ) c o s ³
ñø
L :
òø
( 1 + ) s i n ³
óøy = c o s ³
G r z e g o r z M r z y g Bo c k i , W I L i Z, s e m . I I I , g r . 2 4
x ' = - s i n ³ c o s ³ - ( 1 + c o s ³ ) s i n ³
y ' = - s i n 2 ³ + ( 1 + c o s ³ ) c o s ³
2 2
( x ' ) = s i n 2 ³ c o s 2 ³ + 2 s i n 2 ³ c o s ³ ( 1 + c o s ³ ) + ( 1 + c o s ³ ) s i n 2 ³
2 2
( y ' ) = s i n 4 ³ - 2 ( 1 + c o s ³ ) s i n 2 ³ c o s ³ + ( 1 + c o s ³ ) c o s 2 ³
r ' ( ³ ) = - s i n ³
2 2 2
( x ' ) + ( y ' ) = [ r ' ( ³ ) ] + [ r 2 ( ³ ) ]
2
2 2
( x ' ) + ( y ' ) = s i n 2 ³ + ( 1 + c o s 2 ³ ) = s i n 2 ³ + 1 + 2 c o s ³ + c o s 2 ³ = 2 ( 1 + c o s ³ )
À
À À À
³ ³ ³
ëø öød ³
L = f ( x , y ) d l = 2 ( 1 + c o s ³ ) d ³ = 2 2 c o s 2 ëø öød ³ = 2
ìø ÷ø
+" +" +" +"c o s ìø 2 ÷ø = 4 s i n = 4
2 2
íø øø íø øø
0
L 0 0 0
Z a d . 1 8
1
öø
r ( ³ ) = 3 s i n 3 ëø ³
ìø ÷ø
3
íø øø
ñø 1
öøc o s
( ³ )
ìø
ôøx = 3 s i n 3 ëø ³ ÷ø
3
ôø íø øø
L :
òø
ôøy = 3 s i n 3 ëø 1 ³ öøs i n
( ³ )
ìø ÷ø
ôø
3
íø øø
óø
1 1 1
öø öø öø
x ' = 3 s i n 2 ëø ³ c o s ëø ³ c o s ( ³ ) - 3 s i n 3 ëø ³ s i n ( ³ )
ìø ÷ø ìø ÷ø ìø ÷ø
3 3 3
íø øø íø øø íø øø
1 1 1
öøc o s ëø öø öøc o s
y ' = 3 s i n 2 ëø ³ ³ s i n ( ³ ) + 3 s i n 3 ëø ³ ( ³ )
ìø ÷ø ìø ÷ø ìø ÷ø
3 3 3
íø øø íø øø íø øø
x ' 2 + y ' 2 = r ' 2 ( ³ ) + r 2 ( ³ )
1 1
ëø öøc o s ëø öø
2
r ' ( ³ ) = 3 s i n ³ ³
ìø ÷ø ìø ÷ø
3 3
íø øø íø øø
2 2
îø 1 1 ùø îø 1 ùø 1 îø 1 1 ùø
ëø öøc o s ëø öø ëø öø öø öø ëø öø
2 3
x ' 2 + y ' 2 =
ìø ìø
ïø3 s i n ìø 3 ³ ÷ø ìø 3 ³ ÷øúø + ïø3 s i n ìø 3 ³ ÷øúø = 9 s i n 4 ëø ³ ÷øïøc o s 2 ëø ³ ÷ø + s i n 2 ìø 3 ³ ÷øúø =
3 3
íø øø íø øø íø øø íø øø íø øø íø øø
ðø ûø ðø ûø ðø ûø
1 1
öø öø
= 9 s i n 4 ëø ³ = 3 s i n 2 ëø ³
ìø ÷ø ìø ÷ø
3 3
íø øø íø øø
1
3 À À
"
1 9
ëø öød ³ ³ = ± = 9 2 = À
2
L = f ( x , y ) d l = 3
+" +"s i n ìø 3 ³ ÷ø 3 +"s i n ( ±) d ± 2
íø øø
L 0 0
d ³ = 3 ±
À À À
u = s i n ( ±) v ' = s i n ( ±)
À
2 2
") ( ±) d ± = = [ - s i n ( ±) c o s ( ±) ] + ( ±) d ± =
+"s i n +"s i n ( ±) s i n ( ±) d ± = 0 +"c o s
0 0 0
u ' = c o s ( ±) v = - c o s ( ±)
À À À
À
2 2
= 0 + ( 1 - s i n 2 ( ±) ) d ± = À - ( ±) d ± Ò! ( ±) d ± =
+" +"s i n +"s i n
2
0 0 0
G r z e g o r z M r z y g Bo c k i , W I L i Z, s e m . I I I , g r . 2 5
Z a d . 1 9
x 2 + y 2
A ëø1 , 2 öø B ëø- 1 1 , 1 1 öø Á( x , y ) =
ìø ÷ø ìø ÷ø
3 0
íø øø íø øø
3
ñø
2 = a + b
ñø 3 1 1
ôøa = -
òø1 1 = - 1 1 a + b Ò! ôø 4 Ò! y = - x +
òø
4 4
óø
ôøb = 1 1
ôø
óø 4
2
öø
2 ëø- 3 9
( y ' ) = =
ìø ÷ø
4 1 6
íø øø
2
1 1
ëø öø
1 1 1 3 9 1 2 5 3 3 1 2 1
ëø ëø öød x =
ìø
M = y ) d l = x 2 + - x öø ÷ø 1 + d x = x 2
ìø ÷ø ìø - x +
÷ø
+"Á( x , 3 0 +" +"
ìø ÷ø
4 4 1 6 2 4 1 6 8 1 6
íø øø íø øø
L - 1 1 - 1 1
íø øø
1
2 5 3 3 1 2 1 3 3 3 0 0 3 9 6 0 + 1 4 5 2 3 3 3 0 0 + 1 6 2 3 6 4 9 5 3 6
ëø
= x 3 - x 2 + x öø = + = = = 4 3
ìø ÷ø
1 1 5 2 3 8 4 1 1 5 2 1 1 5 2
íø1 1 5 2 3 8 4 3 8 4 øø
- 1 1
Z a d . 2 0
y = x 0 d" x d" 2 Á( x , y ) = x x
2
öø
2 ëø- 1 1
( y ' ) = =
ìø ÷ø
x 2 x 4
íø øø
2 2 2
1 x x 1 + x 4
M = y ) d l = x x Å" 1 + d x = Å" 1 + x 4 d x = d x =
+"Á( x , +" +" +"
x 4 x 2
x
L 0 0 0
Z a d . 2 1
y = e - 2 x 0 d" x d" 1 Á( x , y ) = y 2
2
2
( y ' ) = ( - 2 e - 2 x ) = 4 e - 4 x
1 + 4 e - 4 x = t
1 1 + 4 e - 4
1
- 4 x
M = y ) d l = 1 + 4 e - 4 x d x = = - t d t =
1
+"Á( x , +"e +"
1 6
e - 4 x d x = - d t
L 0 5
1 6
1 3
îø ùø
= 5 3 - ( 1 + 4 e - 4 )
ïø úø
2 4 ðø ûø
Z a d . 2 2
x = 4 c o s ( 2 t )
ñø
t
ôø
Á ( x , y , ) = 1 +
òøy = 4 s i n ( 2 t ) 0 d" t d" 2 À
2 À
ôøz = 3 t
óø
2 2
2
ñø
( x ' ) = ( - 8 c o s ( 2 t ) s i n ( 2 t ) ) = 6 4 c o s 2 ( 2 t ) s i n ( 2 t )
ôø
ôø 2 2
2
( y ' ) = ( 8 c o s ( 2 t ) s i n ( 2 t ) ) = 6 4 c o s 2 ( 2 t ) s i n ( 2 t )
òø
ôø 2
( z ' ) = 9
ôø
óø
2 À
ëø öø
t
M = y , z ) d l =
+"Á( x , +"ìø1 + 2 À ÷ø 9 + 1 2 8 c o s 2 ( 2 t ) s i n 2 ( 2 t ) d t =
ìø ÷ø
L 0 íø øø
G r z e g o r z M r z y g Bo c k i , W I L i Z, s e m . I I I , g r . 2 6
Z a d . 2 3
y = x 2 0 d" x d" 1
2 2
( y ' ) = ( 2 x ) = 4 x 2
2
1 + 4 x 2 = t
1 5
1 5 3 - 1
M Y = x Á( x , y ) d l = Á0 x d l = Á0 x 1 + 4 x 2 d x = = Á0 2 d t = Á0
1
+" +" +" +"t
4 1 2
x d x = t d t
L L 0 1
4
Z a d . 2 4
x = t - s i n ( t ) y = 1 - c o s ( t ) 0 d" t d" 2 À
2 2
ñø
( x ' ) = ( 1 - c o s ( t ) ) = 1 - 2 c o s ( t ) + c o s 2 ( t )
ôø
òø
2
ôø
( y ' ) = s i n 2 ( t )
óø
2 À 2 À 2 À
M = y ) d l = Á0 1 - 2 c o s ( t ) + c o s 2 ( t ) + s i n 2 ( t ) d t = Á0 2 ( 1 - c o s ( t ) ) d t = 2 Á0 1 - c o s ( t ) d t
+"Á( x , +" +" +"
L 0 0 0
M Y 1 2 Á0 2 À
x C = = x Á( x , y ) d l =
+" +"( t - s i n ( t ) ) 1 - c o s ( t ) d t
M M M
L 0
M 1 2 Á0 2 À
X
y C = = y Á( x , y ) d l =
+" +"( 1 - c o s ( t ) ) 1 - c o s ( t ) d t
M M M
L 0
Z a d . 2 5
x = c o s ( t )
ñø
ôø À
òøy = s i n ( t ) 0 d" t d"
2
ôøz = t
óø
2
ñø
( x ' ) = s i n 2 ( t )
ôø
ôø
2
( y ' ) = c o s 2 ( t )
òø
ôø
2
( z ' ) = 1
ôø
óø
À À
2 2
2 À
M = Á( x , y , z ) d l = Á0 s i n 2 ( t ) + c o s 2 ( t ) + 1 d t = Á0 2 d t = Á0
+" +" +"
2
L 0 0
À
2
M Y Z 1 2 - 2 - 2
x C = = x Á( x , y , z ) d l = Á0
+" +"c o s ( t ) d t = M Á0 = À
M M M
L 0
À
2
M Y Z 1 2 2 2
y C = = x Á( x , y , z ) d l = Á0
+" +"s i n ( t ) d t = M Á0 = À
M M M
L 0
À
2 2 2
M 1 2 À Á0 À Á0 2 À 2
X Y
z C = = z Á( x , y , z ) d l = Á0 = = Å" =
+" +"( t ) d t
M M M 8 M 8 8
2 ÀÁ0
L 0
ëø öø
- 2 2 À 2
÷ø
C ìø , ,
ìø ÷ø
À À 8
íø øø
G r z e g o r z M r z y g Bo c k i , W I L i Z, s e m . I I I , g r . 2 7
Z a d . 2 6
y = e x 0 d" x d" 1
1
I = y , z ) y 2 d l = Á0 2 x 1 + e x d x =
X
+"Á( x , +"e
L 0
Z a d . 2 7
x = c o s ³ z = 0
ñø ñø
òø òø
óøy = s i n ³ óøz = 2 + x
2
( x ' ) = s i n 2 ³
2
( y ' ) = c o s 2 ³
x ' 2 + y ' 2 = ( c o s 2 ³ + s i n 2 ³ = 1
2 À
" = + c o s ³ ) d ³ = 4 À + 0 = 4 À
+"( 2
0
Z a d . 2 8
z = 0
x = 2 c o s ³ ñø
ñø
òø òø
óøy = 2 s i n ³
óøz = 1 + x 2 + y 2
2
( x ' ) = 4 s i n 2 ³
2
( y ' ) = 4 c o s 2 ³
x ' 2 + y ' 2 = 4 ( c o s 2 ³ + s i n 2 ³ = 2
2 À 2 À
" = 4 ) 2 d ³ = = 2 0 À
+"( 1 + +"1 0 d ³
0 0
G r z e g o r z M r z y g Bo c k i , W I L i Z, s e m . I I I , g r . 2 8
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