Маслова Т.Н., Суходский А.М. Справочник школьника по математике. 5-11 кл
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551
550
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
21. Ïðåîáðàçîâàíèå ïðîèçâåäåíèÿ òðèãîíîìåòðè-
÷åñêèõ ôóíêöèé â ñóììó
;
2
)(sin)(sin
cossin
b+a+b-a
=ba
;
2
)
(
cos
)
(
cos
sinsin
b
+a-
b
-a
=
b
a
.
2
)
(
cos
)
(
cos
coscos
b
+a+
b
-a
=
b
a
22. Ðåøåíèå ïðîñòåéøèõ òðèãîíîìåòðè÷åñêèõ
óðàâíåíèé
23. Íåêîòîðûå âàæíûå íåðàâåíñòâà
Íåðàâåíñòâî Êîøè:
,
2
xy
yx
³
+
ãäå
.0,0 ³³ yx
Íåðàâåíñòâî òðåóãîëüíèêà:
.axax +£+
Íåðàâåíñòâî äëÿ ñóììû äâóõ âçàèìíî îáðàòíûõ
ïîëîæèòåëüíûõ âåëè÷èí:
,2
1
³+
x
x
ãäå
.0>x
Íåðàâåíñòâà, õàðàêòåðèçóþùèå ìíîæåñòâî çíà÷å-
íèé ñèíóñà è êîñèíóñà:
.1cos1,1sin1 £a£-£a£-
×àñòíûå ñëó÷àè
0cos =x
1cos =x
Znnx Îp+
p
= ,
2
Znnx Îp= ,2
0tg =x
0ctg =x
1cos -=x
Znnx Îp+p= ,2
Znnx Îp= ,
Znnx Îp+
p
= ,
2
Óðàâíåíèå
Åãî ðåøåíèå
Z
Z
Z
Z
Z
1,cos £= aax
ax =tg
1,sin £= aax
ax =ctg
Znnax
n
Îp+-= ,arcsin)1(
Znnax Îp+= ,arctg
Óðàâíåíèå Åãî ðåøåíèå
x = ± arccos a + 2pn, n Î Z
x = arcctg a + pn, n Î Z
Z
Z
Z
Z
1sin =x
0sin =x
Znnx Îp= ,
Óðàâíåíèå Åãî ðåøåíèå
1sin -=x
Znnx Îp+
p
= ,2
2
Znnx Îp+
p
-= ,2
2
Z
Z
Z
551
550
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
21. Ïðåîáðàçîâàíèå ïðîèçâåäåíèÿ òðèãîíîìåòðè-
÷åñêèõ ôóíêöèé â ñóììó
;
2
)(sin)(sin
cossin
b+a+b-a
=ba
;
2
)
(
cos
)
(
cos
sinsin
b
+a-
b
-a
=
b
a
.
2
)
(
cos
)
(
cos
coscos
b
+a+
b
-a
=
b
a
22. Ðåøåíèå ïðîñòåéøèõ òðèãîíîìåòðè÷åñêèõ
óðàâíåíèé
23. Íåêîòîðûå âàæíûå íåðàâåíñòâà
Íåðàâåíñòâî Êîøè:
,
2
xy
yx
³
+
ãäå
.0,0 ³³ yx
Íåðàâåíñòâî òðåóãîëüíèêà:
.axax +£+
Íåðàâåíñòâî äëÿ ñóììû äâóõ âçàèìíî îáðàòíûõ
ïîëîæèòåëüíûõ âåëè÷èí:
,2
1
³+
x
x
ãäå
.0>x
Íåðàâåíñòâà, õàðàêòåðèçóþùèå ìíîæåñòâî çíà÷å-
íèé ñèíóñà è êîñèíóñà:
.1cos1,1sin1 £a£-£a£-
×àñòíûå ñëó÷àè
0cos =x
1cos =x
Znnx Îp+
p
= ,
2
Znnx Îp= ,2
0tg =x
0ctg =x
1cos -=x
Znnx Îp+p= ,2
Znnx Îp= ,
Znnx Îp+
p
= ,
2
Óðàâíåíèå
Åãî ðåøåíèå
Z
Z
Z
Z
Z
1,cos £= aax
ax =tg
1,sin £= aax
ax =ctg
Znnax
n
Îp+-= ,arcsin)1(
Znnax Îp+= ,arctg
Óðàâíåíèå Åãî ðåøåíèå
x = ± arccos a + 2pn, n Î Z
x = arcctg a + pn, n Î Z
Z
Z
Z
Z
1sin =x
0sin =x
Znnx Îp= ,
Óðàâíåíèå Åãî ðåøåíèå
1sin -=x
Znnx Îp+
p
= ,2
2
Znnx Îp+
p
-= ,2
2
Z
Z
Z
553
552
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
24. Ýëåìåíòû êîìáèíàòîðèêè
Ôîðìóëà ÷èñëà ðàçìåùåíèé áåç ïîâòîðåíèé èç n
ýëåìåíòîâ ïî k ýëåìåíòîâ:
)
,1
)
...
(
1
(
)
!
(
!
+--=
-
= knnn
kn
n
A
k
n
ãäå
.nk £
Ôîðìóëà ÷èñëà ðàçìåùåíèé ñ ïîâòîðåíèÿìè
èç n ýëåìåíòîâ ïî k ýëåìåíòîâ:
.
~
kk
n
nA
=
Ôîðìóëà ÷èñëà ïåðåñòàíîâîê áåç ïîâòîðåíèé èç
n ýëåìåíòîâ:
....321! nnP
n
××==
Ôîðìóëà ÷èñëà ïåðåñòàíîâîê èç n ýëåìåíòîâ ñ
ïîâòîðåíèÿìè, ñîäåðæàùèõ k
1
ýëåìåíòîâ ïåðâîãî òèïà,
k
2
ýëåìåíòîâ âòîðîãî òèïà, ..., k
n
ýëåìåíòîâ
n-ãî òèïà (ãäå
:
)
...
21
nkkk
n
=+++
.
!!...!
)
!...
(
)
...
(
~
21
21
21
n
n
n
kkk
kkk
kkkP
+++
=+++
Ôîðìóëà ÷èñëà ñî÷åòàíèé áåç ïîâòîðåíèé èç
n ýëåìåíòîâ ïî k ýëåìåíòîâ:
,
!)!(
!
kkn
n
P
A
C
k
k
n
k
n
-
==
ãäå
.1,
0
=£
n
Cnk
Ôîðìóëà ÷èñëà ñî÷åòàíèé ñ ïîâòîðåíèÿìè èç
n ýëåìåíòîâ ïî k ýëåìåíòîâ:
.
)
!1
(
!
)
!1
(
~
1
11
-
-+
===
-
-+-+
nk
kn
CCC
n
kn
k
kn
k
n
Ôîðìóëà áèíîìà Íüþòîíà:
...
)
(
22211
++=+
--
baCbaCaba
n
n
n
n
nn
.......
11
nnn
n
kknk
n
babCbaC
++++
---
Ôîðìóëà îáùåãî ÷ëåíà ðàçëîæåíèÿ áèíîìà Íüþ-
òîíà:
,
1
kknk
nk
baCT
-
+
=
ãäå
....,,2,1,0 nk =
25. Àðèôìåòè÷åñêàÿ ïðîãðåññèÿ
Îïðåäåëåíèå àðèôìåòè÷åñêîé ïðîãðåññèè:
.
1
daa
nn
+=
+
Ôîðìóëà n-ãî ÷ëåíà:
).1(
1
-+= ndaa
n
Ôîðìóëû ñóììû n ïåðâûõ ÷ëåíîâ:
;
2
1
n
aa
S
n
n
×
+
=
.
2
)
1
(
2
1
n
nda
S
n
×
-+
=
Õàðàêòåðèñòè÷åñêîå ñâîéñòâî:
.
2
11
+-
+
=
nn
n
aa
a
(a + b)
n
= a
n
+
553
552
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
24. Ýëåìåíòû êîìáèíàòîðèêè
Ôîðìóëà ÷èñëà ðàçìåùåíèé áåç ïîâòîðåíèé èç n
ýëåìåíòîâ ïî k ýëåìåíòîâ:
)
,1
)
...
(
1
(
)
!
(
!
+--=
-
= knnn
kn
n
A
k
n
ãäå
.nk £
Ôîðìóëà ÷èñëà ðàçìåùåíèé ñ ïîâòîðåíèÿìè
èç n ýëåìåíòîâ ïî k ýëåìåíòîâ:
.
~
kk
n
nA
=
Ôîðìóëà ÷èñëà ïåðåñòàíîâîê áåç ïîâòîðåíèé èç
n ýëåìåíòîâ:
....321! nnP
n
××==
Ôîðìóëà ÷èñëà ïåðåñòàíîâîê èç n ýëåìåíòîâ ñ
ïîâòîðåíèÿìè, ñîäåðæàùèõ k
1
ýëåìåíòîâ ïåðâîãî òèïà,
k
2
ýëåìåíòîâ âòîðîãî òèïà, ..., k
n
ýëåìåíòîâ
n-ãî òèïà (ãäå
:
)
...
21
nkkk
n
=+++
.
!!...!
)
!...
(
)
...
(
~
21
21
21
n
n
n
kkk
kkk
kkkP
+++
=+++
Ôîðìóëà ÷èñëà ñî÷åòàíèé áåç ïîâòîðåíèé èç
n ýëåìåíòîâ ïî k ýëåìåíòîâ:
,
!)!(
!
kkn
n
P
A
C
k
k
n
k
n
-
==
ãäå
.1,
0
=£
n
Cnk
Ôîðìóëà ÷èñëà ñî÷åòàíèé ñ ïîâòîðåíèÿìè èç
n ýëåìåíòîâ ïî k ýëåìåíòîâ:
.
)
!1
(
!
)
!1
(
~
1
11
-
-+
===
-
-+-+
nk
kn
CCC
n
kn
k
kn
k
n
Ôîðìóëà áèíîìà Íüþòîíà:
...
)
(
22211
++=+
--
baCbaCaba
n
n
n
n
nn
.......
11
nnn
n
kknk
n
babCbaC
++++
---
Ôîðìóëà îáùåãî ÷ëåíà ðàçëîæåíèÿ áèíîìà Íüþ-
òîíà:
,
1
kknk
nk
baCT
-
+
=
ãäå
....,,2,1,0 nk =
25. Àðèôìåòè÷åñêàÿ ïðîãðåññèÿ
Îïðåäåëåíèå àðèôìåòè÷åñêîé ïðîãðåññèè:
.
1
daa
nn
+=
+
Ôîðìóëà n-ãî ÷ëåíà:
).1(
1
-+= ndaa
n
Ôîðìóëû ñóììû n ïåðâûõ ÷ëåíîâ:
;
2
1
n
aa
S
n
n
×
+
=
.
2
)
1
(
2
1
n
nda
S
n
×
-+
=
Õàðàêòåðèñòè÷åñêîå ñâîéñòâî:
.
2
11
+-
+
=
nn
n
aa
a
(a + b)
n
= a
n
+
555
554
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
26. Ãåîìåòðè÷åñêàÿ ïðîãðåññèÿ
Îïðåäåëåíèå ãåîìåòðè÷åñêîé ïðîãðåññèè:
,
1
qbb
nn
=
+
ãäå
.0,0
1
¹¹ qb
Ôîðìóëà n-ãî ÷ëåíà:
.
1
1
-
=
n
n
qbb
Ôîðìóëû ñóììû n ïåðâûõ ÷ëåíîâ:
;
1
1
-
-
=
q
bqb
S
n
n
.
1
)
1
(
1
-
-
=
q
qb
S
n
n
Õàðàêòåðèñòè÷åñêîå ñâîéñòâî:
.
11
2
+-
×=
nnn
bbb
Ôîðìóëà ñóììû ÷ëåíîâ áåñêîíå÷íîé ãåîìåòðè÷åñ-
êîé ïðîãðåññèè ïðè
:1<q
.
1
1
q
b
S
-
=
27. Ïðîèçâîäíàÿ
Îïðåäåëåíèå ïðîèçâîäíîé:
.
)()(
limlim)(
00
x
xfxxf
x
f
xfy
xx
D
-D+
=
D
D
=
¢
=
¢
®D®D
Òàáëèöà ïðîèçâîäíûõ:
Äèôôåðåíöèðîâàíèå ñóììû, ïðîèçâåäåíèÿ è
÷àñòíîãî:
;)( vuvu
¢
+
¢
=
¢
+
,)( uCCu
¢
=
¢
ãäå Ñ ïîñòîÿííàÿ;
;)( vuvuuv
¢
+
¢
=
¢
,
2
v
vuvu
v
u
¢
-
¢
=
¢
÷
ø
ö
ç
è
æ
ãäå
.0)( ¹xv
Äèôôåðåíöèðîâàíèå ñëîæíîé ôóíêöèè
=y
:))(( xgf=
).())(( xgxgfy
¢¢
=
¢
Óðàâíåíèå êàñàòåëüíîé ê ãðàôèêó ôóíêöèè
)(xfy =
â òî÷êå õ = à:
).()()( axafafy -
¢
+=
5.
.ln)( aaa
xx
=
¢
6.
.
1
)
(
ln
x
x =
¢
7.
.
ln
1
)(log
ax
x
a
=
¢
8.
.cos)(sin xx =
¢
9.
.sin)(cos xx -=
¢
10.
.
cos
1
)
(
tg
2
x
x =
¢
11.
.
sin
1
)
(
ctg
2
x
x -=
¢
12.
.
1
1
)
(
arcsin
2
x
x
-
=
¢
13.
.
1
1
)
(
arccos
2
x
x
-
-=
¢
14.
.
1
1
)
(
arctg
2
x
x
+
=
¢
15.
.
1
1
)
(
arcctg
2
x
x
+
-=
¢
1.
.0=
¢
C
2.
.)( kbkx =
¢
+
3.
.)(
1
-
=
¢
rr
rxx
4.
.)(
xx
ee
=
¢
555
554
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
26. Ãåîìåòðè÷åñêàÿ ïðîãðåññèÿ
Îïðåäåëåíèå ãåîìåòðè÷åñêîé ïðîãðåññèè:
,
1
qbb
nn
=
+
ãäå
.0,0
1
¹¹ qb
Ôîðìóëà n-ãî ÷ëåíà:
.
1
1
-
=
n
n
qbb
Ôîðìóëû ñóììû n ïåðâûõ ÷ëåíîâ:
;
1
1
-
-
=
q
bqb
S
n
n
.
1
)
1
(
1
-
-
=
q
qb
S
n
n
Õàðàêòåðèñòè÷åñêîå ñâîéñòâî:
.
11
2
+-
×=
nnn
bbb
Ôîðìóëà ñóììû ÷ëåíîâ áåñêîíå÷íîé ãåîìåòðè÷åñ-
êîé ïðîãðåññèè ïðè
:1<q
.
1
1
q
b
S
-
=
27. Ïðîèçâîäíàÿ
Îïðåäåëåíèå ïðîèçâîäíîé:
.
)()(
limlim)(
00
x
xfxxf
x
f
xfy
xx
D
-D+
=
D
D
=
¢
=
¢
®D®D
Òàáëèöà ïðîèçâîäíûõ:
Äèôôåðåíöèðîâàíèå ñóììû, ïðîèçâåäåíèÿ è
÷àñòíîãî:
;)( vuvu
¢
+
¢
=
¢
+
,)( uCCu
¢
=
¢
ãäå Ñ ïîñòîÿííàÿ;
;)( vuvuuv
¢
+
¢
=
¢
,
2
v
vuvu
v
u
¢
-
¢
=
¢
÷
ø
ö
ç
è
æ
ãäå
.0)( ¹xv
Äèôôåðåíöèðîâàíèå ñëîæíîé ôóíêöèè
=y
:))(( xgf=
).())(( xgxgfy
¢¢
=
¢
Óðàâíåíèå êàñàòåëüíîé ê ãðàôèêó ôóíêöèè
)(xfy =
â òî÷êå õ = à:
).()()( axafafy -
¢
+=
5.
.ln)( aaa
xx
=
¢
6.
.
1
)
(
ln
x
x =
¢
7.
.
ln
1
)(log
ax
x
a
=
¢
8.
.cos)(sin xx =
¢
9.
.sin)(cos xx -=
¢
10.
.
cos
1
)
(
tg
2
x
x =
¢
11.
.
sin
1
)
(
ctg
2
x
x -=
¢
12.
.
1
1
)
(
arcsin
2
x
x
-
=
¢
13.
.
1
1
)
(
arccos
2
x
x
-
-=
¢
14.
.
1
1
)
(
arctg
2
x
x
+
=
¢
15.
.
1
1
)
(
arcctg
2
x
x
+
-=
¢
1.
.0=
¢
C
2.
.)( kbkx =
¢
+
3.
.)(
1
-
=
¢
rr
rxx
4.
.)(
xx
ee
=
¢
557
556
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
Ôîðìóëà Ëàãðàíæà:
.
)()(
)(
ab
afbf
cf
-
-
=
¢
Ôîðìóëà äëÿ ïðèáëèæåííîãî âû÷èñëåíèÿ çíà÷å-
íèÿ ôóíêöèè
)(xf
â òî÷êå õ:
)
.
(
)
(
)
(
)
(
000
xxxfxfxf -
¢
+»
28. Ïåðâîîáðàçíàÿ è èíòåãðàë
Îïðåäåëåíèå ïåðâîîáðàçíîé: ôóíêöèÿ
)(xF
íà-
çûâàåòñÿ ïåðâîîáðàçíîé äëÿ ôóíêöèè
)(xf
íà ïðî-
ìåæóòêå Õ, åñëè
)()( xfxF =
¢
äëÿ ëþáîãî
.Xx Î
Òàáëèöà ïåðâîîáðàçíûõ:
Ïðàâèëà âû÷èñëåíèÿ ïåðâîîáðàçíûõ:
Åñëè )(xF
ïåðâîîáðàçíàÿ äëÿ
)(xf
, à
)(xH
ïåðâîîáðàçíàÿ äëÿ
),(xh
òî
)()( xHxF +
ïåðâîîá-
ðàçíàÿ äëÿ
).()( xhxf +
Åñëè
)(xF
ïåðâîîáðàçíàÿ äëÿ
),(xf
à k
ïîñòîÿííàÿ, òî
)(xkF
ïåðâîîáðàçíàÿ äëÿ
).(xkf
Åñëè
)(xF
ïåðâîîáðàçíàÿ äëÿ
),(xf
à k è b
ïîñòîÿííûå, ïðè÷åì
,0¹k
òî
)
(
1
bkxF
k
+
ïåðâî-
îáðàçíàÿ äëÿ
).( bkxf +
Èíòåãðàëüíàÿ ñóììà äëÿ ôóíêöèè
)(xfy =
ïî
îòðåçêó
:],[ ba
.)(...)()(
110
-
+++
-
=
nn
xfxfxf
n
ab
S
Èíòåãðàë ôóíêöèè
)(xfy =
îò à äî b:
.lim
)
(
n
n
b
a
Sdxxf
¥®
ò
=
Ôîðìóëà Íüþòîíà Ëåéáíèöà:
.
)
(
)
(
)
(
)
(
b
a
b
a
xFaFbFdxxf
=-
ò
=
x
exf
=)(
x
axf
=)(
a
a
xF
x
ln
)( =
2
1
1
)(
x
xf
-
=
2
1
1
)(
x
xf
+
=
xxF arcsin)( =
xxF arctg)( =
Ôóíêöèÿ Ïåðâîîáðàçíàÿ Ôóíêöèÿ Ïåðâîîáðàçíàÿ
x
xf
2
cos
1
)(
=
r
xxf
=)(
)1( -¹r
kxf =)(
x
xf
1
)( =
kxxF =)(
1
)(
1
+
=
+
r
x
xF
r
xxF ln)( =
x
exF
=)(
xxf sin)( =
xxf cos)( =
x
xf
2
sin
1
)(
=
xxF cos)( -=
xxF sin)( =
xxF ctg)( -=
xxF tg)( =
557
556
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
Ôîðìóëà Ëàãðàíæà:
.
)()(
)(
ab
afbf
cf
-
-
=
¢
Ôîðìóëà äëÿ ïðèáëèæåííîãî âû÷èñëåíèÿ çíà÷å-
íèÿ ôóíêöèè
)(xf
â òî÷êå õ:
)
.
(
)
(
)
(
)
(
000
xxxfxfxf -
¢
+»
28. Ïåðâîîáðàçíàÿ è èíòåãðàë
Îïðåäåëåíèå ïåðâîîáðàçíîé: ôóíêöèÿ
)(xF
íà-
çûâàåòñÿ ïåðâîîáðàçíîé äëÿ ôóíêöèè
)(xf
íà ïðî-
ìåæóòêå Õ, åñëè
)()( xfxF =
¢
äëÿ ëþáîãî
.Xx Î
Òàáëèöà ïåðâîîáðàçíûõ:
Ïðàâèëà âû÷èñëåíèÿ ïåðâîîáðàçíûõ:
Åñëè )(xF
ïåðâîîáðàçíàÿ äëÿ
)(xf
, à
)(xH
ïåðâîîáðàçíàÿ äëÿ
),(xh
òî
)()( xHxF +
ïåðâîîá-
ðàçíàÿ äëÿ
).()( xhxf +
Åñëè
)(xF
ïåðâîîáðàçíàÿ äëÿ
),(xf
à k
ïîñòîÿííàÿ, òî
)(xkF
ïåðâîîáðàçíàÿ äëÿ
).(xkf
Åñëè
)(xF
ïåðâîîáðàçíàÿ äëÿ
),(xf
à k è b
ïîñòîÿííûå, ïðè÷åì
,0¹k
òî
)
(
1
bkxF
k
+
ïåðâî-
îáðàçíàÿ äëÿ
).( bkxf +
Èíòåãðàëüíàÿ ñóììà äëÿ ôóíêöèè
)(xfy =
ïî
îòðåçêó
:],[ ba
.)(...)()(
110
-
+++
-
=
nn
xfxfxf
n
ab
S
Èíòåãðàë ôóíêöèè
)(xfy =
îò à äî b:
.lim
)
(
n
n
b
a
Sdxxf
¥®
ò
=
Ôîðìóëà Íüþòîíà Ëåéáíèöà:
.
)
(
)
(
)
(
)
(
b
a
b
a
xFaFbFdxxf
=-
ò
=
x
exf
=)(
x
axf
=)(
a
a
xF
x
ln
)( =
2
1
1
)(
x
xf
-
=
2
1
1
)(
x
xf
+
=
xxF arcsin)( =
xxF arctg)( =
Ôóíêöèÿ Ïåðâîîáðàçíàÿ Ôóíêöèÿ Ïåðâîîáðàçíàÿ
x
xf
2
cos
1
)(
=
r
xxf
=)(
)1( -¹r
kxf =)(
x
xf
1
)( =
kxxF =)(
1
)(
1
+
=
+
r
x
xF
r
xxF ln)( =
x
exF
=)(
xxf sin)( =
xxf cos)( =
x
xf
2
sin
1
)(
=
xxF cos)( -=
xxF sin)( =
xxF ctg)( -=
xxF tg)( =
559
558
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
Ïðàâèëà âû÷èñëåíèÿ èíòåãðàëîâ:
òò
+
ò
=+
b
a
b
a
b
a
dxxfdxxfdxxfxf
.
)
(
)
(
))
(
)
(
(
2121
òò
=
b
a
b
a
dxxfkdxxkf
,
)
(
)
(
ãäå k ïîñòîÿííàÿ.
Ôîðìóëà äëÿ âû÷èñëåíèÿ ïëîùàäè ôèãóðû, îã-
ðàíè÷åííîé ïðÿìûìè õ = à, x = b (a < b) è ãðàôèêà-
ìè íåïðåðûâíûõ íà [a, b] ôóíêöèé
),(
1
xfy =
=y
),(
2
xf=
ãäå
)()(
12
xfxf £
íà [a, b]:
.
))
(
)
(
(
21
ò
-=
b
a
dxxfxfS
Ôîðìóëà äëÿ âû÷èñëåíèÿ ïëîùàäè êðèâîëèíåéíîé
òðàïåöèè, ò. å. ôèãóðû, îãðàíè÷åííîé ïðÿìûìè ó =
= 0, õ = à, õ = b (a < b) è ãðàôèêîì íåïðåðûâíîé è
íåîòðèöàòåëüíîé íà [a, b] ôóíêöèè
)(xfy =
:
.
)
(
ò
=
b
a
dxxfS
Ôîðìóëà äëÿ âû÷èñëåíèÿ îáúåìà òåëà, ðàñïî-
ëîæåííîãî ìåæäó äâóìÿ ïëîñêîñòÿìè, óäàëåííû-
ìè îò äàííîé ïëîñêîñòè íà ðàññòîÿíèÿ à è b (a < b)
è òàêîãî, ÷òî â ïåðåñå÷åíèè ýòîãî òåëà ïëîñêîñòüþ,
ïàðàëëåëüíîé äàííîé è óäàëåííîé îò íåå íà ðàñ-
ñòîÿíèå
),( bxax <<
îáðàçóåòñÿ ñå÷åíèå, ïëîùàäü
êîòîðîãî âûðàæàåòñÿ íåïðåðûâíîé íà [a, b] ôóíê-
öèåé S (x):
ò
=
b
a
dxxSV
.
)
(
Ôîðìóëà äëÿ âû÷èñëåíèÿ îáúåìà òåëà, îáðàçîâàí-
íîãî âðàùåíèåì âîêðóã îñè àáñöèññ êðèâîëèíåéíîé
òðàïåöèè, îãðàíè÷åííîé ïðÿìûìè ó = 0, õ = à, õ =
= b (a < b) è ãðàôèêîì íåïðåðûâíîé íà [a, b] ôóíê-
öèè
)(xfy =
:
.
2
ò
p=
b
a
dxyV
;
559
558
ÎÑÍÎÂÍÛÅ ÔÎÐÌÓËÛ Àëãåáðà
Ïðàâèëà âû÷èñëåíèÿ èíòåãðàëîâ:
òò
+
ò
=+
b
a
b
a
b
a
dxxfdxxfdxxfxf
.
)
(
)
(
))
(
)
(
(
2121
òò
=
b
a
b
a
dxxfkdxxkf
,
)
(
)
(
ãäå k ïîñòîÿííàÿ.
Ôîðìóëà äëÿ âû÷èñëåíèÿ ïëîùàäè ôèãóðû, îã-
ðàíè÷åííîé ïðÿìûìè õ = à, x = b (a < b) è ãðàôèêà-
ìè íåïðåðûâíûõ íà [a, b] ôóíêöèé
),(
1
xfy =
=y
),(
2
xf=
ãäå
)()(
12
xfxf £
íà [a, b]:
.
))
(
)
(
(
21
ò
-=
b
a
dxxfxfS
Ôîðìóëà äëÿ âû÷èñëåíèÿ ïëîùàäè êðèâîëèíåéíîé
òðàïåöèè, ò. å. ôèãóðû, îãðàíè÷åííîé ïðÿìûìè ó =
= 0, õ = à, õ = b (a < b) è ãðàôèêîì íåïðåðûâíîé è
íåîòðèöàòåëüíîé íà [a, b] ôóíêöèè
)(xfy =
:
.
)
(
ò
=
b
a
dxxfS
Ôîðìóëà äëÿ âû÷èñëåíèÿ îáúåìà òåëà, ðàñïî-
ëîæåííîãî ìåæäó äâóìÿ ïëîñêîñòÿìè, óäàëåííû-
ìè îò äàííîé ïëîñêîñòè íà ðàññòîÿíèÿ à è b (a < b)
è òàêîãî, ÷òî â ïåðåñå÷åíèè ýòîãî òåëà ïëîñêîñòüþ,
ïàðàëëåëüíîé äàííîé è óäàëåííîé îò íåå íà ðàñ-
ñòîÿíèå
),( bxax <<
îáðàçóåòñÿ ñå÷åíèå, ïëîùàäü
êîòîðîãî âûðàæàåòñÿ íåïðåðûâíîé íà [a, b] ôóíê-
öèåé S (x):
ò
=
b
a
dxxSV
.
)
(
Ôîðìóëà äëÿ âû÷èñëåíèÿ îáúåìà òåëà, îáðàçîâàí-
íîãî âðàùåíèåì âîêðóã îñè àáñöèññ êðèâîëèíåéíîé
òðàïåöèè, îãðàíè÷åííîé ïðÿìûìè ó = 0, õ = à, õ =
= b (a < b) è ãðàôèêîì íåïðåðûâíîé íà [a, b] ôóíê-
öèè
)(xfy =
:
.
2
ò
p=
b
a
dxyV
;