1 Definicja [a, b] n n " N
P = {x0, x1, . . . , xn},
a = x0 < x1 < · · · < xn = b
2 Definicja f [a, b] P
f P ¾k
¾ " [xk-1, xk] 1 d" k d" n
n

S = f(¾k)"xk, "xk = xk - xk-1.
k=1
3 Przykład f(x) = 3 [1, 2] P
n ¾k [xk-1, xk]
n n

S = f(¾k) · (xk - xk-1) = 3(xk - xk-1) = 3(xn - x0) = 3(2 - 1) = 3.
k=1 k=1
f(x) = x [1, 2] P
n ¾k = xk-1
n
n n

2 - 1 xk-1
k=1
S = f(xk-1) · (xk - xk-1) = xk-1 · =
n n
k=1 k=1

n n
(0+n-1)·n
1 1 1
x0 + (k - 1) · nx0 + (k - 1)
k=1 k=1
n n n 2
= = = x0 +
n n n
n - 1
= 1 + .
2n
f(x) = x [1, 2] P
n ¾k = xk
n
n n

2 - 1 xk
k=1
S = f(xk) · (xk - xk-1) = xk · =
n n
k=1 k=1

n
(1+n)·n
k 1
x0 + nx0 + · n + 1
k=1
n n 2
= = = 1 + .
n n 2n
4 Definicja f [a, b]
f [a, b]

n
b

f(x) dx = lim f(¾k)"xk,
´(P)0
a
k=1

´(P ) = max {"xk}
1d"kd"n
P ¾k


a a b
f(x) dx = 0 f(x) dx = - f(x) dx a < b.
a b a
5 Przykład f(x) = x [1, 2] Pn n

lim ´(P ) = 0.
n"
¾k = xk-1

n - 1 1 3
lim S = lim S = lim 1 + = 1 + = .
n" n"
´(P)0
2n 2 2
¾k = xk

n + 1 1 3
lim S = lim S = lim 1 + = 1 + = .
n" n"
´(P)0
2n 2 2

¾k

2
3
x dx = .
2
1
6 Definicja f
f

7 TWIERDZENIE f [a, b]

8 Przykład f(x) = x [1, 2]
¾k

9 TWIERDZENIE f [a, b]

b
f(x) dx = F (b) - F (a),
a
F f
10 Przykład F G f C
F (x) = G(x) + C x
F (b) - F (a) = [G(b) + C] - [G(a) + C] = G(b) + C - G(a) - C = G(b) - G(a).

11 TWIERDZENIE a d" b d" c


c b b
f(x) dx = f(x) dx + f(x) dx.
a a b
12 Przykład

3
3x2 dx = x3|2 = 33 - 13 = 27 - 1 = 26,
1
1
4
3x2 dx = x3|4 = 43 - 33 = 256 - 27 = 229,
3
3
4
3x2 dx = x3|4 = 43 - 13 = 256 - 1 = 255 = 26 + 229.
1
1
13 TWIERDZENIE


b b b
[f(x) + g(x)]dx = f(x) dx + g(x) dx
a a a


b b
[a · f(x)]dx = a · f(x) dx.
a a
14 Przykład
2

2
x3 23 03 8
(x2 + 7)dx = + 7x = + 7 · 2 - - 7 · 0 = + 14
3 3 3 3
0
0
2

2 2

23 03 x3

= - + 7 · 2 - 7 · 0 = + 7x|2 = x2dx + 7 dx
0
3 3 3
0 0
0
15 TWIERDZENIE u v [a, b]


b b
u(x)v2 (x) dx = [u(x)v(x)]b - u2 (x)v(x) dx.
a
a a
16 Przykład


Ä„/2 Ä„/2

u = x u2 = 1

x sin x dx = = [-x cos x]Ä„/2 + cos x dx
0

v2 = sin x v = - cos x
0 0
= 0 + [sin x]Ä„/2 = 1.
0
17 TWIERDZENIE
g [a, b]
f [g(a), g(b)]



b g(b)
f (g(x)) g2 (x) dx = f(t) dt.
a g(a)
18 Przykład
1

Ä„/2 1
t3 1
t = sin x
sin2 x cos x dx = = t2 dt = = .
dt = cos x dx
3 3
0 0
0



19 TWIERDZENIE f [a, b] g
f g
[a, b]

b b
g(x) dx = f(x) dx.
a a

20 TWIERDZENIE [a, b] f(x) e" 0
y = f(x) X x = a x = b

b
f(x dx).
a
21 Przykład f(x) = x + 1 x = 1
x = 2
x " [1, 2] f(x) > 0
2

2

x2 22 12 5
A = (x + 1)dx = + x = + 2 - - 1 = .

2 2 2 2
1
1
2 x = 1 3 x = 2
1
(2 + 3) · 1 5
A = = .
2 2
22 TWIERDZENIE f g [a, b] f(x) d" g(x)
x " [a, b] f g
x = a x = b

b
P = [g(x) - f(x)] dx.
a

23 Przykład
24 TWIERDZENIE f [a, b]

x, f(x) : x " [a, b]


b

L = 1 + |f2 (x)|2 dx.
a
25 Przykład
26 TWIERDZENIE f [a, b] T
f OX x = a x = b

T OX


b
V = Ä„ f2(x) dx,
a
T OY


b
V = 2Ä„ xf(x) dx.
a
27 Przykład
28 TWIERDZENIE f [a, b]
f OX

b

P = 2Ä„ f(x) 1 + |f2 (x)|2 dx,
a
f OY

b

P = 2Ä„ x 1 + |f2 (x)|2 dx.
a
29 Przykład

30 Definicja f [a, ")
f [a, ")

" T
f(x) dx = lim f(x) dx.
T "
a a

f (-", b] f
(-", b]

b b
f(x) dx = lim f(x) dx.
T -"
-" T
f (-", ")
f (-", ")

" a "
f(x) dx = f(x) dx + f(x) dx,
-" -" a
a
Ä…"
-" +"

31 Definicja f (a, b]
a f (a, b]


b b
f(x) dx = lim f(x) dx.
%EÅ‚0+ a+%EÅ‚
a
f [a, b)
b f [a, b)

b b-%EÅ‚
f(x) dx = lim f(x) dx.
%EÅ‚0+ a
a
f [a, c) *" (c, b]
c f [a, b]

b c b
f(x) dx = f(x) dx + f(x) dx.
a a c