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8.5. ²íòåãðóâàííÿ äåÿêèõ ³ððàö³îíàëüíèõ<br />
ôóíêö³é ....................................................... 280<br />
8.5.1. ²íòåãðàëè âèäó Rxx ( , α β<br />
∫ , x, K ) dx ..................... 280<br />
8.5.2. ²íòåãðàëè ç ðàäèêàëàìè ............................... 280<br />
Âïðàâè .................................................................. 281<br />
8.6. ²íòåãðóâàííÿ âèðàç³â ç òðèãîíîìåòðè÷íèìè<br />
ôóíêö³ÿìè ............................................. 282<br />
Âïðàâè .................................................................. 284<br />
ÒÅÌÀ 9. ÂÈÇÍÀ×ÅÍÈÉ ²ÍÒÅÃÐÀË<br />
9.1. Çàäà÷³, ÿê³ ïðèâîäÿòü äî ïîíÿòòÿ âèçíà-<br />
÷åíîãî ³íòåãðàëà .......................................... 286<br />
9.1.1. Çàäà÷à ïðî ïëîùó êðèâîë³í³éíî¿<br />
òðàïåö³¿ ....................................................... 286<br />
9.1.2. Çàäà÷à ïðî ðîáîòó çì³ííî¿ ñèëè .................. 289<br />
9.1.3. Çàäà÷à ïðî îáñÿã ïðîäóêö³¿ .......................... 291<br />
9.2. Îçíà÷åííÿ âèçíà÷åíîãî ³íòåãðàëà ................. 291<br />
9.2.1. Îçíà÷åííÿ òà óìîâè ³ñíóâàííÿ âèçíà-<br />
÷åíîãî ³íòåãðàëà .......................................... 292<br />
9.2.2. Ãåîìåòðè÷íèé, ìåõàí³÷íèé ³ åêîíîì³÷íèé<br />
çì³ñò âèçíà÷åíîãî ³íòåãðàëà .................. 293<br />
9.3. Âëàñòèâîñò³ âèçíà÷åíîãî ³íòåãðàëà ................ 295<br />
9.4. Îñíîâíà ôîðìóëà ³íòåãðàëüíîãî ÷èñëåííÿ ..... 300<br />
9.4.1. ²íòåãðàë ³ç çì³ííîþ âåðõíüîþ ìåæåþ .......... 300<br />
9.4.2. Ôîðìóëà Íüþòîíà — Ëåéáí³öà .................... 301<br />
9.5. Îñíîâí³ ìåòîäè îá÷èñëåííÿ âèçíà÷åíèõ<br />
³íòåãðàë³â .................................................... 302<br />
9.5.1. Áåïîñåðåäí³é ìåòîä ...................................... 302<br />
9.5.2. Ìåòîä çàì³íè çì³ííî¿, àáî ï³äñòàíîâêè ......... 303<br />
9.5.3. Ìåòîä ³íòåãðóâàííÿ ÷àñòèíàìè ..................... 306<br />
Âïðàâè .................................................................. 307<br />
9.6. Íåâëàñí³ ³íòåãðàëè ....................................... 307<br />
9.6.1. Íåâëàñí³ ³íòåãðàëè ² ðîäó ........................... 307<br />
9.6.2. Íåâëàñí³ ³íòåãðàëè II ðîäó .......................... 312<br />
Âïðàâè .................................................................. 317<br />
9.7. Ãåîìåòðè÷í³ çàñòîñóâàííÿ âèçíà÷åíîãî<br />
³íòåãðàëà ..................................................... 317<br />
9.7.1. Îá÷èñëåííÿ ïëîù ô³ãóð ............................... 317<br />
Âïðàâè .................................................................. 323<br />
9.7.2. Îá÷èñëåííÿ äîâæèíè äóã êðèâèõ<br />
ë³í³é .......................................................... 323<br />
Âïðàâè .................................................................. 327<br />
9.8. Îá÷èñëåííÿ îá’ºì³â ò³ë .............................. 327<br />
9.8.1. Îá÷èñëåííÿ îá’ºì³â ò³ë çà â³äîìèìè<br />
ïëîùàìè ïåðåð³ç³â ..................................... 327<br />
9.8.2. Îá’ºì ò³ëà îáåðòàííÿ ................................ 329<br />
Âïðàâè .................................................................. 330<br />
9.9. Ïëîùà ïîâåðõí³ ò³ëà îáåðòàííÿ .................. 330<br />
9.9.1. Âèïàäîê îáåðòàííÿ êðèâî¿, ÿêà çàäàíà<br />
ð³âíÿííÿì y = f(x), x∈ [a, b] ........................ 330<br />
9.9.2. Âèïàäîê îáåðòàííÿ êðèâî¿, ÿêà çàäàíà<br />
ïàðàìåòðè÷íî ............................................. 331<br />
Âïðàâè .................................................................. 332<br />
9.10. Ô³çè÷í³ çàñòîñóâàííÿ âèçíà÷åíîãî<br />
³íòåãðàëà.................................................... 332<br />
9.10.1. Ðîáîòà çì³ííî¿ ñèëè .................................. 332<br />
9.10.2. Îá÷èñëåííÿ ïðîéäåíîãî øëÿõó ................... 334<br />
9.11. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà<br />
â åêîíîì³ö³ ................................................ 335<br />
9.11.1. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà<br />
â äèíàì³÷íèõ ïðîöåñàõ .............................. 335<br />
9.11.2. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà<br />
â çàäà÷àõ ðåàë³çàö³¿ òîâàð³â ....................... 337<br />
9.11.3. Çàñòîñóâàííÿ âèçíà÷åíîãî ³íòåãðàëà<br />
â çàäà÷àõ îá÷èñëåííÿ âèòðàò, äîõîä³â<br />
òà ïðèáóòê³â .............................................. 339<br />
9.12. Íàáëèæåíå îá÷èñëåííÿ âèçíà÷åíèõ<br />
³íòåãðàë³â .................................................. 340<br />
9.12.1. Ôîðìóëà ïðÿìîêóòíèê³â ............................. 340<br />
9.12.2. Ôîðìóëà òðàïåö³é ...................................... 341<br />
ÒÅÌÀ 10. ÔÓÍÊÖ²¯ ÁÀÃÀÒÜÎÕ Ç̲ÍÍÈÕ<br />
10.1. Îñíîâí³ îçíà÷åííÿ ³ ïîíÿòòÿ ...................... 343<br />
10.1.1. Ïðîáëåìí³ ïðèêëàäè .................................. 343<br />
10.1.2. Îçíà÷åííÿ ôóíêö³¿ áàãàòüîõ çì³ííèõ.......... 344<br />
10.1.3. Ôóíêö³ÿ äâîõ çì³ííèõ ³ ¿¿ îáëàñòü<br />
âèçíà÷åííÿ................................................. 345<br />
Âïðàâè .................................................................. 351<br />
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