# integrale

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```							‫)ﻣﺤﻤﺪ اﻟﻜﯿﺎل(‬                                   ‫اﻟﺘﻜﺎﻣﻞ‬

‫‪Ë‬ﺗﻜﺎﻣﻞ داﻟﺔ ﻣﺘﺼﻠﺔ ﻋﻠﻰ ﻗﻄﻌﺔ:‬
‫‪ f‬داﻟﺔ ﻣﺘﺼﻠﺔ ﻋﻠﻰ ﻣﺠﺎل ]_ ‪ [ !a, b‬و ‪ F‬داﻟﺔ أﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪ f‬ﻋﻠﻰ اﻟﻤﺠﺎل ]_ ‪[ !a, b‬‬                  ‫ﻟﺘﻜﻦ‬
‫ﺗﻜﺎﻣﻞ اﻟﺪاﻟﺔ ‪ f‬ﻣﻦ ‪ a‬إﻟﻰ ‪ b‬ھﻮ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ:‬
‫‪b‬‬                           ‫‪b‬‬
‫) ‪òa f ( x )dx = éF ( x )ùa = F ( b ) - F ( a‬‬
‫‪ë‬‬       ‫‪û‬‬

‫‪Ë‬ﺧﺎﺻﯿﺎت:‬
‫اﻟﺨﻄﺎﻧﯿﺔ:‬
‫‪a‬‬                ‫‪b‬‬                                                   ‫‪a‬‬
‫‪òb‬‬  ‫‪f ( x )dx = - ò f ( x )dx‬‬                                        ‫0 = ‪òa f ( x )dx‬‬
‫‪a‬‬
‫‪b‬‬                           ‫‪b‬‬             ‫‪b‬‬                                        ‫‪b‬‬                 ‫‪b‬‬
‫‪éf ( x ) + g ( x ) ùdx = ò f ( x )dx + ò g ( x )dx‬‬
‫‪òa ë‬‬                  ‫‪û‬‬                                               ‫)¡ ‪(k Î‬‬     ‫‪òa‬‬  ‫‪kf ( x )dx = k ò f ( x )dx‬‬
‫‪a‬‬             ‫‪a‬‬                                                          ‫‪a‬‬
‫ﻋﻼﻗﺔ ﺷﺎل:‬
‫‪b‬‬              ‫‪c‬‬             ‫‪b‬‬
‫‪òa‬‬ ‫‪f ( x )dx = ò f ( x )dx + ò f ( x )dx‬‬
‫‪a‬‬             ‫‪c‬‬
‫‪Ë‬اﻟﺘﻜﺎﻣﻞ و اﻟﺘﺮﺗﯿﺐ:‬
‫] ‪"x Î [ a, b‬‬        ‫إذا ﻛﺎن: ) ‪f ( x ) £ g ( x‬‬                     ‫] ‪"x Î [ a, b‬‬      ‫إذا ﻛﺎن: 0 ³ ) ‪f ( x‬‬
‫‪b‬‬                  ‫‪b‬‬                                                      ‫‪b‬‬
‫‪òa‬‬    ‫ﻓﺈن: ‪f ( x )dx £ ò g ( x )dx‬‬                                          ‫ﻓﺈن: 0 ³ ‪òa f ( x )dx‬‬
‫‪a‬‬
‫‪Ë‬اﻟﻘﯿﻤﺔ اﻟﻤﺘﻮﺳﻄﺔ:‬
‫‪ f‬داﻟﺔ ﻣﺘﺼﻠﺔ ﻋﻠﻰ ﻣﺠﺎل ]‪[a,b‬‬        ‫ﻟﺘﻜﻦ‬
‫1‬   ‫‪b‬‬
‫‪b-a‬‬ ‫اﻟﻘﯿﻤﺔ اﻟﻤﺘﻮﺳﻄﺔ ﻟﻠﺪاﻟﺔ ﻋﻠﻰ اﻟﻤﺠﺎل ]‪ [a,b‬ھﻲ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ: ‪òa f ( x )dx‬‬
‫‪Ë‬اﻟﻤﻜﺎﻣﻠﺔ ﺑﺎﻷﺟﺰاء:‬
‫‪ [ #a, b‬ﺑﺤﯿﺚ ﺗﻜﻮن '‪ f‬و'‪ g‬ﻣﺘﺼﻠﺘﯿﻦ‬            ‫]‬   ‫ﻟﺘﻜﻦ ‪ f‬و ‪ g‬داﻟﺘﯿﻦ ﻗﺎﺑﻠﺘﯿﻦ ﻟﻼﺷﺘﻘﺎق ﻋﻠﻰ ﻣﺠﺎل‬
‫ﻋﻠﻰ اﻟﻤﺠﺎل ] ‪[ #a, b‬‬
‫‪b‬‬                                                     ‫‪b‬‬
‫‪éf ' ( x ) ´ g ( x ) ù dx = éf ( x ) ´ g ( x ) ù - ò éf ( x ) ´ g ' ( x ) ùdx‬‬
‫‪b‬‬
‫‪òa‬‬ ‫‪ë‬‬                    ‫‪û‬‬      ‫‪ë‬‬                  ‫‪ûa‬‬   ‫‪aë‬‬                    ‫‪û‬‬
‫‪Ë‬ﺣﺴﺎب ﻣﺴﺎﺣﺔ ﺣﯿﺰ:‬
‫‪rr‬‬
‫(‬            ‫)‬
‫ﻟﯿﻜﻦ اﻟﻤﺴﺘﻮى ﻣﻨﺴﻮﺑﺎ إﻟﻰ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ‪o,i, j‬‬
‫وﺣﺪة اﻟﻤﺴﺎﺣﺔ : ‪ u . A‬ھﻲ ﻣﺴﺎﺣﺔ اﻟﻤﺴﺘﻄﯿﻞ اﻟﻤﺤﺪد‬
‫‪r r‬‬
‫ﺑﺎﻟﻨﻘﻄﺔ ‪ o‬و اﻟﻤﺘﺠﮫﺘﯿﻦ ‪ i‬و ‪j‬‬
‫‪r‬‬       ‫‪r‬‬
‫‪1.u . A = i × j‬‬

‫22‬
‫‪ g‬داﻟﺘﯿﻦ ﻣﺘﺼﻠﺘﯿﻦ ﻋﻠﻰ ﻣﺠﺎل ]‪[a;b‬‬                    ‫ﻟﺘﻜﻦ ‪ f‬و‬                    ‫‪ f‬داﻟﺔ ﻣﺘﺼﻠﺔ ﻋﻠﻰ ﻣﺠﺎل ]‪[a;b‬‬     ‫ﻟﺘﻜﻦ‬
‫ﻣﺴﺎﺣﺔ اﻟﺤﯿﺰ اﻟﻤﺤﺼﻮر ﺑﯿﻦ اﻟﻤﻨﺤﻨﯿﯿﻦ ‪ Cf‬و‬                          ‫ﻣﺴﺎﺣﺔ اﻟﺤﯿﺰ اﻟﻤﺤﺼﻮر ﺑﯿﻦ اﻟﻤﻨﺤﻨﻰ ‪ Cf‬وﻣﺤـﻮر‬
‫‪ Cg‬وﻣﺤﻮر اﻷﻓﺎﺻﯿﻞ واﻟﻤﺴﺘﻘﯿﻤﯿﻦ اﻟﻠﺬﻳﻦ‬                              ‫اﻷﻓﺎﺻﯿﻞ واﻟﻤﺴﺘﻘﯿﻤﯿﻦ اﻟﻠﺬﻳﻦ ﻣﻌﺎدﻟﺘﺎھﻤﺎ:‬
‫‪ x = a‬و ‪ y = b‬ھﻲ:‬
‫ﻣﻌﺎدﻟﺘﺎھﻤﺎ: ‪ x = a‬و ‪ y = b‬ھﻲ:‬
‫‪æ b‬‬             ‫‪ö‬‬
‫‪æ b‬‬
‫‪ç òa‬‬
‫‪ö‬‬
‫‪f ( x ) - g ( x ) dx ÷ .u.A‬‬                                       ‫‪ç òa f ( x ) dx ÷ .u.A‬‬
‫‪è‬‬                         ‫‪ø‬‬                                            ‫‪è‬‬               ‫‪ø‬‬

‫ﺣﺎﻻت ﺧﺎﺻﺔ:‬

‫ﻣﺴﺎﺣﺔ اﻟﺤﯿﺰ اﻟﺮﻣﺎدي ﻓﻲ اﻟﺮﺳﻢ‬                                      ‫ﻣﻼﺣﻈﺎت‬                            ‫اﻟﺮﺳﻢ‬

‫‪æ b‬‬            ‫‪ö‬‬                                     ‫‪ f‬ﻣﻮﺟﺒﺔ‬
‫‪ç òa f ( x )dx ÷ .u.A‬‬                            ‫ﻋﻠﻰ اﻟﻤﺠﺎل ]‪[a,b‬‬
‫‪è‬‬              ‫‪ø‬‬

‫‪æ b‬‬             ‫‪ö‬‬                                      ‫‪ f‬ﺳﺎﻟﺒﺔ‬
‫‪ç òa -f ( x )dx ÷ .u.A‬‬                             ‫ﻋﻠﻰ اﻟﻤﺠﺎل ]‪[a,b‬‬
‫‪è‬‬               ‫‪ø‬‬

‫· ‪ f‬ﻣﻮﺟﺒﺔ‬
‫‪æ c‬‬               ‫‪b‬‬            ‫‪ö‬‬                               ‫ﻋﻠﻰ اﻟﻤﺠﺎل ]‪[a,c‬‬
‫‪ç òa f ( x )dx + òc -f ( x )dx ÷ .u.A‬‬                                   ‫· ‪ f‬ﺳﺎﻟﺒﺔ‬
‫‪è‬‬                              ‫‪ø‬‬
‫ﻋﻠﻰ اﻟﻤﺠﺎل ]‪[c, b‬‬

‫‪æ b‬‬                          ‫‪ö‬‬                          ‫) ‪ ( Cf‬ﻳﻮﺟﺪ ﻓﻮق ) ‪( Cg‬‬
‫‪ç òa ( f ( x ) - g ( x ) )dx ÷ .u.A‬‬
‫‪è‬‬                            ‫‪ø‬‬                             ‫ﻋﻠﻰ اﻟﻤﺠﺎل ]‪[a,b‬‬
‫) ‪ ( Cf‬ﻓﻮق ) ‪( Cg‬‬      ‫·‬
‫‪æ c‬‬                             ‫‪b‬‬                         ‫‪ö‬‬
‫‪ç òa ( f ( x ) - g ( x ) )dx + òc ( g ( x ) - f ( x ) )dx ÷ .u.A‬‬      ‫ﻋﻠﻰ اﻟﻤﺠﺎل ]‪[a,c‬‬
‫‪è‬‬                                                         ‫‪ø‬‬
‫· ) ‪ ( Cg‬ﻓﻮق ) ‪( Cf‬‬
‫ﻋﻠﻰ اﻟﻤﺠﺎل ]‪[c, b‬‬
‫‪Ë‬ﺣﺴﺎب ﺣﺠﻢ :‬
‫‪ ( Cf‬ﺣﻮل‬   ‫)‬    ‫ﺣﺠﻢ اﻟﻤﺠﺴﻢ اﻟﻤﻮﻟﺪ ﺑﺪوران اﻟﻤﻨﺤﻨﻰ‬
‫ﻣﺤﻮر اﻷﻓﺎﺻﯿﻞ دورة ﻛﺎﻣﻠﺔ ﻓﻲ ﻣﺠﺎل ]‪[a,b‬‬
‫‪é b‬‬              ‫‪² ù‬‬
‫‪V = ê ò p ( f ( x ) ) dx ú u.v‬‬    ‫ھﻮ:‬
‫‪ë a‬‬                  ‫‪û‬‬

‫32‬

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