[ a, b ] n n " N P =
{x0, x1, . . . , xn} a = x0 < x1 < � � � < xn = b "xk
k "xk = xk - xk-1 �(P ) = max "xk
1d"kd"n
P x" k
k
x" " [kk-1, xk]
k
f [ a, b ] f
P [ a, b ] n x", x", . . . , x"
1 2 n
n
S(f, P ) = S(f, P, x", x", . . . , x") = f(x")"xk.
1 2 n k
k=1
y = f(x) x = a x = b
"xk (x")
k
f [ a, b ]
f [ a, b ]
n
b
f(x) dx = lim f(x")"xk
k
�(P )0
a
k=1
 [ a, b ] x", x", . . . , x"
1 2 n
a a b
f(x) dx = 0 oraz f(x) dx = - f(x) dx (dla a < b).
a b a
[ a, b ]
[a, b]
[ a, b ]
b b
f(x) dx, f(x) dx, f, f.
a a
[a,b] [a,b]
f
b
f(x) dx
a
[ a, b ] �" R
1, x
f(x) =
0, x
f
[ a, b ]
f [ a, b ]
 f, g : [a, b ] R
[ a, b ] C " R a < d < b
f
C � f f ą g f � g g(x) = 0 x " [a, b ]

g
b b
C � f(x) dx = C � f(x) dx
a a
b b b
(f(x) ą g(x)) dx = f(x) dx ą g(x) dx.
a a a
|f|
b b
f(x) dx d" |f(x)| dx,
a a
d b b
f(x) dx + f(x) dx = f(x) dx
a d a
b b
f(x) d" g(x) f(x) dx d" g(x) dx
a a
f [ a, b ]
b
b
f(x) dx = F (b) - F (a) = F (x) ,
a
a
F f
f [ a, b ]
b
n
b - a b - a
f(x) dx = lim f a + k � .
n"
n n
k=1
a
 f g [ a, b ]
b b
b
f(x)g (x) dx = f(x)g(x) - f (x)g(x) dx.
a
a a
f : [ a, b ] R � : [ą, �] [a, b]
[ą, �] �(ą) = a �(�) = b
b �
f(x) dx = f(�(t))� (t) dt.
a ą
f [ a, b ] f [ a, b ]
b
1
f = f(x) dx.
b - a
a
f [ a, b ]
b-a
f Ox x = a x = b
f R
[ a, b ] �" R
f f(-x) = -f(x) x " R
a > 0
a
f(x) dx = 0,
-a
 f f(-x) = f(x) x " R
a > 0
a a
f(x) dx = 2 f(x) dx,
-a 0
f T f(x + kT ) = f(x) k " Z
a " R
a+T T
f(x) dx = f(x) dx
a 0
f [ a, b ] D
f Ox x = a x = b
|D| "Dk
�(P ) 0
b
n n
|D| = lim |"Dk| = lim f(x")"xk = f(x) dx.
k
�(P )0 �(P )0
k=1 k=1
a
f Ox f
x " [ a, b ]
 S [ą, �]
v(t) t " [ą, �] S
"Sk "tk
v(t") �(P ) 0
k
�
n n
S = lim |"Sk| = lim v(t")"tk = v(t) dt.
k
�(P )0 �(P )0
k=1 k=1
ą
S
v Ot t = ą t = �
F (t) = (x(t), y(t)) [ a, b ]
x y [ a, b ] F
�ł �ł
b b b
�ł
F (t) dt = x(t) dt, y(t) dtłł .
a a a
F (t) = (x(t), y(t), z(t))
d, g : [ a, b ] R d(x) d" g(x)
x " [ a, b ] D
d g x = a x = b
b
|D| = (g(x) - d(x)) dx.
a
 d, g : [ p, q ] R d(y) d" g(y)
y " [ p, q ] D
d g y = p y = q
q
|D| = (g(y) - d(y)) dy.
p
f [ a, b ] f C1 [ a, b ]
� = {(x, f(x)) : x " [ a, b ]}
b
|�| = 1 + (f (x))2 dx.
a
S(x) x " [ a, b ] V
Ox x S [ a, b ]
V
b
|V | = S(x) dx.
a
T
[ a, b ] Ox x = a x = b
V T Ox
b
|V | = Ą f2(x) dx.
a
 V T Oy
a > 0
b
|V | = 2Ą xf(x) dx.
a
f [ a, b ]
Ł f
� = {(x, f(x)) : x " [ a, b ]} Ox
b
|Ł| = 2Ą f(x) 1 + (f (x))2 dx.
a
a e" 0 Ł
f � = {(x, f(x)) : x " [ a, b ]} Oy
b
|Ł| = 2Ą x 1 + (f (x))2 dx.
a
f : [ a, ") R f
[ a, ")
" T
f(x) dx = lim f(x) dx.
T "
a a
T "
lim f(x) dx f(x) dx
T "
a a
T "
lim f(x) dx ą" f(x) dx
T "
a a
ą"
T "
lim f(x) dx f(x) dx
T "
a a
(-", b ]
b b
f(x) dx = lim f(x) dx.
T -"
-" T
f : (-", ") R
f (-", ")
" a "
f(x) dx = f(x) dx + f(x) dx,
-" -" a
a
a " "
f(x) dx f(x) dx f(x) dx
-" a -"
a "
f(x) dx " f(x) dx
-" a
"
" f(x) dx "
-"
a "
f(x) dx " f(x) dx
-" a
"
-" f(x) dx
-"
a
f(x) dx
-"
" "
f(x) dx f(x) dx
a -"
"
f(x) dx a " R
-"
a " R a
a > 0
"
1
dx
xp
a
p > 1 p d" 1
b
1
dx b < 0
xp
-"
 ( x, y) x, y " R R2
R2 = {(x, y) : x, y " R}.
( x, y, z) x, y, z " R R3
R3 = {(x, y, z) : x, y, z " R}.
x, y, z
R2 P1 = (x1, y1) P2 = (x2, y2)
|P1P2| = (x1 - x2)2 + (y1 - y2)2.
R3 P1 = (x1, y1, z1) P2 = (x2, y2, z2)
|P1P2| = (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2.
r > 0 P0 R2 R3
O(P0, r) = {P : |P P0| < r}.
P0 r
A
P0 r > 0 A �" O(P0, r)
A
 A
P " A r > 0 O(P, r) �" A
A
A
P A
P A A
A Ac r > 0
O(P, r) )" A = " '" O(P, r) )" Ac = ".

P A
A
A "A
f A �" R2
R A
f : A R z = f(x, y) (x, y) " A
f(x, y)
f (x, y)
f Df
f
f
f(x, y) = 4 - x2 - y2
4 - x2 - y2 e" 0
Df = {(x, y) " R2 : x2 + y2 d" 4}
(0, 0) 2
 f
{(x, y, z) " R3 : (x, y) " Df, z = f(x, y)}.
f h
{(x, y) " Df : f(x, y) = h}.
z = Ax + By + C
n = (-A, -B, 1) P = (0, 0, C)
z = a(x2 + y2)
z = ax2 Oz
z = a x2 + y2
z = a |x| Oz
z = g( x2 + y2)
z = g(|x|) Oz
z = R - (x2 + y2)
(0, 0, 0) R z = - R - (x2 + y2)
z = g(x) f(x, y) = g(x) y " R
z = g(x)
y = 0 Oy
z = f(x-a, y -b)+c
z = f(x, y) v = (a, b, c)
z = -f(x, y) z =
f(x, y) xOy z = 0