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calcule d'integrale

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									                                                                                      ‫ــــــــــ ــــــــــ ب ا ـــــــــــ ـــــــــــ ـــــــــــ ــــــــــ‬
                                                                                                                : ‫ه‬           - (6                                                   :             ‫دا‬                                              :         - (1
   . y0 ∈ R ‫ و‬a ∈ I                                       ‫ و‬I ‫ل‬                                             ‫ دا‬f                                        [a       ; b] ‫ل‬          ‫ا‬               f ‫ا‬                                  ‫ دا أ‬F
                                x
         G ( x) = ∫ f ( x)dx :                                       ‫آ‬I                            ‫ا‬G            ‫*** ا ا‬                      b

                                                                                                                                              ∫   f ( x)dx = [ f ( x) ]a = F (b) − F (a)
                                                                                                                                                                                             b
                             a

   . x0 = a                 ‫م‬                     ‫ ا‬I                 f               ‫ا‬               ‫ا ا ا‬                                   a
            x                                                                                                                                                                                                                                 :‫ت‬             - (2
 F ( x) = ∫ f ( x)dx + y0 :                                         ‫ آ‬I                            ‫ ا‬F           ‫*** ا ا‬                                 a                               b                                a

            a                                                                                                                                           ∫ f ( x)dx = 0 ; ∫ f ( x)dx = −∫ f ( x)dx                                                           ***
 . x0 = a               ‫ا‬             y0                  ‫ا‬                   ‫ا‬f          ‫ا‬                 ‫ه ا ا ا‬                                      a                               a                                b
                                                                                                                                                        b                    c                                c
                                                                                                :‫ت‬               ‫با‬           - (7
                                    [a        ; b] ‫ل‬                                               ‫ دا‬f                   ***                           ∫ f ( x)dx + ∫ f ( x)dx = ∫ f ( x)dx
                                                                                                                                                        a                    b                                a
                                                                                                                                                                                                                                                            ***


         (O ; i ; j )                                                     f           ‫ا ا‬             (C )   f                  ‫و‬
                                                                                                                                                        b                                        b

                                                                                                                                                        ∫ ( f (t ) + g (t ) ) dt = ∫ f (t )dt + ∫ g (t )dt
                                                                                                                                                                                                                                  b



     ‫( و‬C f     )               ‫ر‬                 ‫∆ ا‬f ‫ي‬                          ‫ا‬                          ‫ا‬                   ‫ا‬                      a                                        a                                a
                                                                                                                                                                                                                                                            ***
                                                                                                                                                        b                            b
                : ‫ ه‬x=b ‫ و‬x=a                                                                 ‫وا‬                   ‫را‬
                                          b
                                                                                                                                                        ∫ λ f (t )dt = λ ∫ f (t )dt
                                                                                                                                                        a                            a

         A(∆ f ) = ∫ f ( x) dx ( u.a)                                                                                                                                                                b

                                          b
                                                                                                                                                        a ≤ b ; f ( x) ≥ 0 ⇒ ∫ f ( x) dx ≥ 0                                                                ***

(O ; i ; j )
                                                                                                                                                                                                     a
                                ‫ا‬             ‫ا‬           ‫ت‬                   ‫سا‬              ‫ ه و ة‬u.a                                                                                      b                            b
                                                                                                                                                        a ≤ b ; f ≤ g ⇒ ∫ f ( x)dx ≤ ∫ g ( x)dx                                                             ***
                                              u.a = i × j                                                        :              ‫و‬                                                            a                            a

                        [a           ; b]                      ‫ن‬                  ‫ دا ن‬g ‫ و‬f                              ***
                                                                                                                                                                                                                              :                     ‫ا‬       ‫3( - ا‬
                                                                                                                                                             :         [ a ; b] ‫ل‬                                                             ‫ دا‬f
         ‫ ( و‬Cg ) ‫ ( و‬C f                         )             ‫ي‬                 ‫ا‬                          ‫ا‬                   ‫ا‬
                                                                                                                                                                     f ([ a; b ]) = [ m; M ]
                                                               : ‫ ه‬x=b ‫ و‬x=a                                                     ‫ا‬                                                                        b
                    b                                                                                                                                                                         1
                                                                                                                                                                                             b−a∫
                                                                                                                                                                                 m≤               f ( x)dx ≤ M :
        S = ∫ f ( x) − g ( x) dx (ua )                                                                                                                                                          a
                    a                                                                                                                                                                                             b
                                                                                                                                                                                      1
                                                                                                                                                                 ∃c ∈ [ a; b ] ; µ =
                                                                                                                                                                                     b−a ∫
                                                                                                        :                 ***                                                              f ( x)dx = f (c) ‫و‬
                            R                                                                                                                                                            a
            S = 2∫                    R 2 − x 2 dx
                            −R
                                                                    : ‫ ه‬R                             ‫ص‬                                 [a    ; b] ‫ل‬             ‫ا‬       f       ‫ا‬                            ‫ا‬           ‫ا‬                             µ
                                                                                                                                                                                                                          :‫اء‬                               ‫4( - ا‬
                = π R2                                                                                                                    b                                                                           b

                                                                                                                                         ∫ u( x)v '( x)dx = [u( x)v( x)] − ∫ u '( x)v( x)dx
                                                                                                   :‫م‬           ‫با‬            - (8                                                                        b
            [a          ; b] ‫ل‬                                                ‫و‬                    ‫ دا‬f                   ***
                                                                                                                                          a
                                                                                                                                                                                                          a
                                                                                                                                                                                                                      a
      ‫ دورة آ‬f                      ‫ا ا‬                       ‫وران‬                    ‫ا‬       ‫ا ورا‬                   ‫ا‬                                                                                               :               ‫ا‬                      ‫5( ا‬
                        b
                                                                                                                                                   g ‫ [ و‬a ; b] ‫ل‬
                                                                                                                                                    '
                                                                                                                                                                                         ‫ا‬           ‫ق‬                                             ‫ دا‬g
            V = ∫ π ( f ( x) ) dx (u.v) : ‫ه‬
                                                      2
                                                                                                        ‫را‬                ‫ل‬
                        a                                                                                                                               g ( [ a; b ] )                                   ‫ دا‬f ‫ [ و‬a ; b ] ‫ل‬                                     ‫ا‬
                                                          2
                 u.v = i × j                                   :‫م‬             ‫سا‬              ‫ ه و ة‬u.v                               x = g (t ) ⇔ dx = g '(t )dt               b                                                        g (b )
                                                                                                                                     
                                                                R                         ‫آ ة‬               :             ***        t = a ⇔ x = g (a)                          ∫ f ( g (t )) g '(t )dt = ∫                                       f ( x)dx
                                                                                                                                     t = b ⇔ x = g (b)
                                              (                               ) dx = 4 π R (uv)
                                R                                                 2                                                                                              a                                                        g (a)
                                                                                                                                     
               V =              ∫π                    R2 − x2                                                    3

                             −R
                                                                                     3

								
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