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# 02 intégrale sur un segment d une fonction continue par morceaux

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### 02 intégrale sur un segment d une fonction continue par morceaux

1. 1. ! " # ! < \$ % & ! ' & ! [ ]# \$#"""#( ) * =<<<= """) " % \$#"""#( )=σ \$+#"""+#+(+ )=σ & ! [ ]# # * +σ * σ * { } { }+#"""+#+#"""# )) ⊂ % \$#"""#( )=σ \$+#"""+#+(+ )=σ & ! * * [ ]# # * , & * & ! * +σ * σ & { } { }+#"""+#+#"""# )) ∪ " -\$ . ' * [ ]# [ ]# & ! \$#"""#( )=σ [ ]# * * ! ! ] [#− # " / & ! σ & " ' ! * # σ & ! & # & ! * σ & " [ ]# * & ! # & " 0 [ ]# # & ! & ! σ +σ # & ! ++σ * σ +σ & " * & # * # [ ]# # & ! & ++σ " 1 # * & [ ]# (* [ ]# # \$ 2& 3 2& [ ]# ( ! \$" touscours.net
2. 2. \$ % [ ]# # \$#"""#( )=σ & ! & " 4 # [ ]# # ! ! ! ] [#− " ! & = −−= \$(\$#( σ & ! σ & " ' & * +σ σ , # \$#(\$+#( σσ = " 1 # & , * +σ * σ # \$#(\$+#( σσ = " 0 # # & ! ++σ * +σ σ # \$#(\$++#(\$+#( σσσ == " ' [ ]# ! \$#( σ # & ! σ & " [ ]# " # ! [ ] = −−= # \$( " ! 5 * 6 & * & " ' * * [ ]# \$( − 7 ! " ' * ! ( [ ]# # λ µ \$ 3 ! )≥ [ ]# # [ ] ) # ≥ 3 / [ ] [ ] [ ] +=+ ### µλµλ 3 ! % << # [ ] [ ] [ ] += ### ( ! * & [ ]# [ ]# \$" 3 % ≤ [ ]# # [ ] [ ] ≤ ## ( ! \$ % [ ]# # * & " ' " % \$(− ε & ϕ [ ]# * ( + 3 3 * ≤ϕ \$ touscours.net
3. 3. 8 % \$(+ ε & ψ [ ]# * ( + 3 3 * ψ≤ \$ / & \$(− ε \$(+ ε ! \$(− ε # \$(+ ε " % \$(− & \$(− ε " % \$(+ & \$(+ ε " / & \$(− \$(+ & ! # \$(− \$(+ # ϕ ψ [ ]# * ψϕ ≤≤ # ψϕ ≤ # [ ] [ ] ≤ ## ψϕ " 1 \$(− & # \$(− # \$(+ & # \$(+ " # \$(\$( +− ≤ " % & # * & [ ]# # [ ]# ! & " 1 # & [ ]# # * & [ ]# " & & 9 " ' * # # & & [ ]# # * : ! ( # ! * # * # \$(− ε \$(+ ε ; \$ 0 # & [ ]# # [ ]# [ ]# # \$( ( 2 # ! & # * \$" ' ! * & ! 5 * 6 & * & " \$ 1 % [ ]# " ' * [ ]# & ! \$#"""#( )=σ [ ]# * * # ! ! ] [#− # − " / & ! σ & " ' ! * [ ]# # & ! σ & # & ! * σ & " * [ ]# * & (! ; \$# * touscours.net
4. 4. < ( \$" % [ ]# # \$#"""#( ) & ! & " # * [ ]# # ! ! ] [#− & [ ]#− " * [ ]# & # ! # * [ ]# # [ ]#− # & " & # & [ ]# # , [ ] [ ] [ ]− ### ) ) #"""##\$(#"""\$(#\$( ' # * & [ ]# [ ]# " 1 # * [ ]# 2& 3 2& [ ]# " -\$ 0 2 % [ ]# " # ε # ϕ [ ]# * εϕ ≤− " 1 • = 7 [ ]# " % )>ε [ ]# ( 2 > \$# )>α * [ ] [ ] ( )εα <−<−∈∀∈∀ \$+(\$(+##+## ' 2 * α< − # & ! 2 \$#"""#( )=σ [ ]# [ ] +=∈∀ ##) # ! − = ( & ! 2 σ \$ ' 2 ϕ [ ] [ [ \$(\$(#### −− =∈∀∈∀ ϕ \$(\$( =ϕ " εϕ ≤− % [ ]#∈ 3 % = # εϕ ≤=− )\$(\$( 3 % # [ ]#∈ * [ [#−∈ " touscours.net
5. 5. ? εϕ <−=− − \$(\$(\$(\$( # 2 ! * # [ [#−∈ # α<−≤− −− • % * ' & ! \$#"""#( )=σ & # # * [ ]# # 2 \$# * [ ]#− * : ! ] [#− " 0 )>ε # * * # * [ ]#− ϕ * [ ] εϕ ≤−∈∀ − \$(\$(## " ' ϕ [ ]# [ ] \$(\$(##) =∈∀ ϕ # [ ] ] [ \$(\$(#### ϕϕ =∈∀∈∀ − " ϕ ! [ ]# # εϕ ≤− " 2 # 2 (! \$ % [ ]# " # ε # ϕ ψ [ ]# * ψϕ ≤≤ εϕψ ≤− / ! @ # ϕ * εϕ ≤− # εϕϕ −=+ εϕψ +=+ ++ ψϕ ≤≤ εϕψ ≤− # ! # ψϕ ≤≤ εϕψ ≤− # ! εϕ ≤− " \$ * & 2 [ ]# & [ ]# " 1 % [ ]# " % )>ε " % 2 # ϕ ψ [ ]# * ψϕ ≤≤ εϕψ ≤− " # # \$(− ∈εϕ \$(+ ∈εψ # [ ] [ ] ≤≤≤ +− ## \$(\$( ψϕ " # [ ] [ ] [ ] [ ] εεψϕψϕ \$( #### −=≤−=− 1 ε\$(\$(\$() −≤−≤ −+ ! & * * )>ε # # # * )\$(\$( =− −+ # # & [ ]# " touscours.net
6. 6. ' * [ ]# * # ε # ϕ ψ [ ]# * ψϕ ≤≤ εϕψ ≤− " & # * & " 0 # * * 2 5 [ ]# 6# # 2 # ! * 2 5 [ ]# & 6" ' & & " A * • 0 & & " % * [ ]#) ( + 3 3 \$( = ∈ # ) \$ % ϕ [ ]#) ≤ϕ # ! ] [#− & ! & ϕ # ! ϕ ) * ! ) ] [#− (* \$" 1 )\$( =− " 1 @ # \$( =+ # 7 & ( 9 \$" • 0 # @ # & % [ ]#) \$( = )≠ # ) " 1 ,# * )" 1 # ! ϕ ψ [ ]#) * ψϕ ≤≤ ≤−ϕψ " 0 # ! 3 ! [ ]α#) ! ) ≤< α # ϕ ψ ! [ ]#) & ! & ϕ ψ # ! " # ε * ) << ε " [ ]#ε # & # ! # ϕ ψ [ ]#ε # ε " % ϕ ψ [ ]#) \$( −=ϕ \$( =ψ [ [ε#)∈ # * ϕ ψ [ ]#) # * ψϕ ≤≤ # * [ ] [ ] [ ] [ ] εεεεϕψϕψϕψ εε 8\$( ##)#)#) ≤−+≤−+−=− 1 7# 2 )\$(\$( =− −+ " • 0 # * " % [ ]#) [ ] \$( = )≠ # ) " # ( { })B∈ \$" # * ε 2 (! * C ( * ε< \$# [ ]#) ) # ! ] ]# \$ # # & " ( −= π \$" touscours.net