"
"
"
"
"
 [a, b] n n " N
P = {x0, x1, . . . , xn} ,
a = x0 < x1 < · · · < xn = b.
1 k n
"xk := xk - xk-1 k P
´(P) := max {"xk} P
x" " [xk-1, xk] k P
k
f [a, b] P
f
P x"
k
n

Ã(f, P) := f(x")"xk.
k
k=1
f [a, b]
f [a, b]
b
n

f(x)dx := lim f(x")"xk,
k
´(P)0
k=1
a
P
x"
k
[a, b]
[a, b]
[a, b] f(x) 0
y = f(x)
b
f(x)dx.
a
 [a, b] f(x) 0 y = f(x)
b
- f(x)dx.
a
b a a
f(x)dx = - f(x)dx f(x)dx = 0.
a b a
f [a, b]
b
f(x)dx = [F (x)]b = F (b) - F (a),
a
a
F f
2
e-xdx
-1
e
dx
x
1
f g [a, b]
b b b

f(x) + g(x) dx = f(x)dx + g(x)dx;
a a a
b b

cf(x) dx = c f(x)dx, x " R.
a a
f [a, b] c " [a, b]
b c b
f(x)dx = f(x)dx + f(x)dx.
a a c
 3
|x - 2| dx.
-1
Õ : [Ä…, ²] [a, b] [Ä…, ²]
Õ(Ä…) = a, Õ(²) = b
f [a, b]
b ²

f(x)dx = f Õ(t) Õ (t)dt.
a Ä…
2 1
2 xdx
xex dx "
5 - 4x
0 -1
f g [a, b]
b
b b
f(x)g (x)dx = f(x)g(x) - f (x)g(x)dx.
a
a a
e
ln2 xdx
1
"
3
2

arcsin xdx
1
-
2
x = -1 x = 1
1
y = .
x2 + 1
y2 = 2x
x = 8
 y = x3 +
x2 - 2x x = -2, x = 2
y = x2 + 6x - 7 y = 1 - x

"
Ä„ Ä„
y = 3 cos x y = sin x - x
2 2
f [a, b]
“ = {(x, f(x)) : x " [a, b]}
b

|“| = 1 + [f (x)]2dx.
a
r
Ä„
y = ln cos x 0 x
3
"
1
y = 1 - x2 0 x
2
f [a, b]
y = f(x) x " [a, b]
OX
b
V = Ä„ [f(x)]2dx.
a
R
OX
1
y = 0 x 1
1
"+ x2
y = xe-x 0 x 4
f [a, b]
 y = f(x)
x " [a, b] OX
b

S = 2Ä„ f(x) 1 + [f (x)]2dx.
a
" OX
y = x 0 x 1
y = sin x 0 x Ä„
c " [a, b] f
µ b
lim f(x)dx lim f(x)dx,
µc- µc+
a µ
f
[a, b]
b
f(x)dx.
a
[a, b]
b µ
lim f(x)dx lim f(x)dx.
µa+ µb-
µ a
1
xdx
"
1 - x2
0
1
dx
"
1 - x2 arcsin x
1
2
 f [a, "]
µ
lim f(x)dx,
µ"
a
f [a, "]
"
f(x)dx.
a
b
f(x)dx
-"
b
lim f(x)dx.
µ-"
µ
"
e-xdx
0
2
"
2 1
+ dx
x x2
1
"
(arctan x)2dx
1 + x2
-"