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4.1 INTÉGRALE CURVILIGNE

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Texte intégral

(1)

cours 25

4.1 INTÉGRALE

CURVILIGNE

(2)

Aujourd’hui, nous allons voir

(3)

Aujourd’hui, nous allons voir

Intégrale curviligne de champs scalaire

(4)

Aujourd’hui, nous allons voir

Intégrale curviligne de champs scalaire Champs vectoriels conservatifs

(5)

Aujourd’hui, nous allons voir

Intégrale curviligne de champs scalaire Champs vectoriels conservatifs

Intégrale curviligne de champs de vecteurs

(6)

C

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(7)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

(8)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

~r(s)

<latexit sha1_base64="8+i13ibFNLg3ScJaIt+MmvxTXtw=">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</latexit>

(9)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

~r(s)

<latexit sha1_base64="8+i13ibFNLg3ScJaIt+MmvxTXtw=">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</latexit>

(10)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

~r(s)

<latexit sha1_base64="8+i13ibFNLg3ScJaIt+MmvxTXtw=">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</latexit>

(11)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

~r(s)

<latexit sha1_base64="8+i13ibFNLg3ScJaIt+MmvxTXtw=">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</latexit>

(12)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

~r(s)

<latexit sha1_base64="8+i13ibFNLg3ScJaIt+MmvxTXtw=">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</latexit>

Xn

k=1

f (xk, yk) sk

<latexit sha1_base64="OxFBz6UtJsKkjPsh9o4dKD5lKI8=">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</latexit>

(13)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

~r(s)

<latexit sha1_base64="8+i13ibFNLg3ScJaIt+MmvxTXtw=">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</latexit>

Xn

k=1

f (xk, yk) sk

<latexit sha1_base64="OxFBz6UtJsKkjPsh9o4dKD5lKI8=">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</latexit>

(14)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

~r(s)

<latexit sha1_base64="8+i13ibFNLg3ScJaIt+MmvxTXtw=">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</latexit>

Xn

k=1

f (xk, yk) sk

<latexit sha1_base64="OxFBz6UtJsKkjPsh9o4dKD5lKI8=">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</latexit>

nlim!1

Xn

k=1

f (xk, yk) sk = Z

C

f (x, y)ds

<latexit sha1_base64="vhsjJUKdTvywH9GUAwPZDUz+VSM=">AAADSnicjVJNT9VAFL0UVMQPnrh0M/HFBAx5adFENyREXLjEyAMSCs103jyYdPqRzpTQNO+H+R/8A67cys64M248MxSjEqPTtHPm3HPu3DudtNLK2DD8OBfML9y4eWvx9tKdu/fuLw8erOyZsqmFHItSl/VByo3UqpBjq6yWB1UteZ5quZ9m2y6+fyZro8pi17aVPMr5SaGmSnALKhm8i7XKk66IbcliVUxtO4tNAybbjGbHBZuunifZ8dN11rppLX4tteXMJNmmk9uki3NuTwXXbHsG7Xq7NjHJYBiOQj/YdRD1YEj92CkHXyimCZUkqKGcJBVkgTVxMngOKaKQKnBH1IGrgZSPS5rRErwNVBIKDjbD9wSrw54tsHY5jXcL7KLx1nAyegJPCV0N7HZjPt74zI79W+7O53S1tZjTPlcO1tIp2H/5rpT/63O9WJrSS9+DQk+VZ1x3os/S+FNxlbNfurLIUIFzeIJ4DSy88+qcmfcY37s7W+7jn73SsW4tem1DF32Vks581vZn9Z3/hwrxyp9lC2Tx9X8JVyL68wJcB3sbo+jZaOPt8+HWq/5yLNIjekyruAEvaIve0A6NUc17+oQKL4IPwdfgW/D9UhrM9Z6H9NuYX/gBc8i7HQ==</latexit>

(15)

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit> ~r(t)

<latexit sha1_base64="WLrztjHuw1KmXRzAetfvWs9H8hU=">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</latexit>

~r(s)

<latexit sha1_base64="8+i13ibFNLg3ScJaIt+MmvxTXtw=">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</latexit>

Xn

k=1

f (xk, yk) sk

<latexit sha1_base64="OxFBz6UtJsKkjPsh9o4dKD5lKI8=">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</latexit>

nlim!1

Xn

k=1

f (xk, yk) sk = Z

C

f (x, y)ds

<latexit sha1_base64="vhsjJUKdTvywH9GUAwPZDUz+VSM=">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</latexit>

(16)

Z

C

f (x, y)ds = Z

C

f (x, y) ds

dt dt

<latexit sha1_base64="XxsGEd3cHTYhRTuwmwWyeAx55qY=">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</latexit>

(17)

Z

C

f (x, y)ds = Z

C

f (x, y) ds

dt dt

<latexit sha1_base64="XxsGEd3cHTYhRTuwmwWyeAx55qY=">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</latexit>

= Z

C

f (x, y)k~r 0(t)kdt

<latexit sha1_base64="OwFDSKCfCSidgGIMgDbcIcX7shs=">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</latexit>

(18)

Z

C

f (x, y)ds = Z

C

f (x, y) ds

dt dt

<latexit sha1_base64="XxsGEd3cHTYhRTuwmwWyeAx55qY=">AAADOXicjVJNTxRBEH2MqIioKx69TNiYQGI2s2CiFxMCF4+YuEDCkk1PTy90mK909xAnk/0//gf/A1f16A09mfgHrC4aohKiPZme16/q1VRVV1rn2rok+TIX3Zq/fefuwr3F+0sPHj7qPV7etVVjpBrJKq/MfiqsynWpRk67XO3XRokizdVeerLt7Xunylhdle9cW6vDQhyVeqqlcERNelvxWJdu0o0L4Y6lyOPt2XT1/fN2LbOvbzKNp0bILrOzLnOzzE16/WSQ8Iqvg2EAfYS1U/V+YIwMFSQaFFAo4QjnELD0HGCIBDVxh+iIM4Q02xVmWCRtQ16KPASxJ7Qf0ekgsCWdfUzLakl/yek1pIzxjDQV+RnC/m8x2xuO7NmbYncc0+fW0jcNsQpiHY6J/Zfu0vN/db4WhylecQ2aaqqZ8dXJEKXhrvjM49+qchShJs7jjOyGsGTlZZ9j1liu3fdWsP2cPT3rzzL4NvgWslQ45ajtVfYd36Eme829bAk52vmWaCSGfw/AdbC7PhhuDNbfvuhvboXhWMBTrGCVJuAlNvEGOxhRNh9whk/4HH2Mvkbn0fcL12guaJ7gjxX9/AVLMLYc</latexit>

= Z

C

f (x, y)k~r 0(t)kdt

<latexit sha1_base64="OwFDSKCfCSidgGIMgDbcIcX7shs=">AAADInicjVLNThRBEP4YVBD/Fjx66bgxrtFsZpEELiZELh4xcYGEIZue3l7oMH/p6dkwWXkI38F38KpXb8aTiQ+gNx6B6rIxKjHak+n5+qv6aqqqK60yU7s4/jIXzV+5em1h8frSjZu3bt/pLK/s1GVjlR6qMivtXiprnZlCD51xmd6rrJZ5mund9HjL23en2tamLF65ttIHuTwszMQo6YgadR4/E4kp3GiW5NIdKZmJrdNJ7+RJ+yh5nUy1EjYRD3uOTmM36nTjfsxLXAaDALoIa7vsnCHBGCUUGuTQKOAIZ5Co6dnHADEq4g4wI84SMmzXOMUSaRvy0uQhiT2m/ZBO+4Et6Oxj1qxW9JeMXktKgQekKcnPEvZ/E2xvOLJn/xZ7xjF9bi190xArJ9bhiNh/6S48/1fna3GYYINrMFRTxYyvToUoDXfFZy5+qcpRhIo4j8dkt4QVKy/6LFhTc+2+t5Lt39jTs/6sgm+D7yFLjSlHbX9mP+M7NGSvuJctIUc73xKNxODPAbgMdlb7g6f91Zdr3c3nYTgWcQ/30aMJWMcmXmAbQ8rmDd7hPT5Eb6OP0afo8w/XaC5o7uK3FX09BxWxq4E=</latexit>

=

Z b

a

f x(t), y(t) k~r 0(t)kdt

<latexit sha1_base64="LFwS0TrU+5lCUyce1kkPD/ffgug=">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</latexit>

(19)

Z

C

f (x, y)ds = Z

C

f (x, y) ds

dt dt

<latexit sha1_base64="XxsGEd3cHTYhRTuwmwWyeAx55qY=">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</latexit>

= Z

C

f (x, y)k~r 0(t)kdt

<latexit sha1_base64="OwFDSKCfCSidgGIMgDbcIcX7shs=">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</latexit>

=

Z b

a

f x(t), y(t) k~r 0(t)kdt

<latexit sha1_base64="LFwS0TrU+5lCUyce1kkPD/ffgug=">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</latexit>

=

Z b

a

f x(t), y(t) p

x0(t)2 + y0(t)2dt

<latexit sha1_base64="tYFaDhp2nmX4MvvDxjr4NCiNiEY=">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</latexit>

(20)

Z

C

f (x, y)ds = Z

C

f (x, y) ds

dt dt

<latexit sha1_base64="XxsGEd3cHTYhRTuwmwWyeAx55qY=">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</latexit>

= Z

C

f (x, y)k~r 0(t)kdt

<latexit sha1_base64="OwFDSKCfCSidgGIMgDbcIcX7shs=">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</latexit>

=

Z b

a

f x(t), y(t) k~r 0(t)kdt

<latexit sha1_base64="LFwS0TrU+5lCUyce1kkPD/ffgug=">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</latexit>

=

Z b

a

f (~r(t))k~r 0(t)kdt

<latexit sha1_base64="HZZTp24lK4R2GUvagD1IMnldlRA=">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</latexit>

=

Z b

a

f x(t), y(t) p

x0(t)2 + y0(t)2dt

<latexit sha1_base64="tYFaDhp2nmX4MvvDxjr4NCiNiEY=">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</latexit>

(21)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

(22)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

~r<latexit sha1_base64="N6i6loCT/ZfyJ/EXnCDPT35FFgE=">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</latexit> (t) = (2 cos t, 2 sin t)

(23)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">AAADFXicjVJNaxRBEH0Zv2L8WvUoyOAieNBldhX0GMzFYwQ3CWSTpae3kzSZL2Z6gsOSW/5D/oNXvXoTr579B3rzJ/i67IRoEO1hel6/qldTVV1pldnGJcnXhejCxUuXryxeXbp2/cbNW73bd9aasq21GesyK+uNVDUms4UZO+sys1HVRuVpZtbT/RVvXz8wdWPL4o3rKrOVq93C7litHKlp7/7EFm46n+TK7WmVxSuHb7dHT7rtUTx5HM+aaa+fDBJZ8XkwDKCPsFbL3g9MMEMJjRY5DAo44gwKDZ9NDJGgIreFObmayIrd4BBL1Lb0MvRQZPe57/K0GdiCZx+zEbXmXzK+NZUxHlJT0q8m9n+Lxd5KZM/+LfZcYvrcOn7TECsn67BH9l+6E8//1flaHHbwQmqwrKkSxlenQ5RWuuIzj89U5RihIufxjPaaWIvypM+xaBqp3fdWif2beHrWn3XwbfE9ZGlwIFG70+zncoeW9kp62RE57nJLHInhnwNwHqyNBsOng9HrZ/3ll2E4FnEPD/CIE/Acy3iFVYyZzRHe4T0+RMfRx+hT9PmXa7QQNHfx24q+/AQgRaaP</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

~r<latexit sha1_base64="N6i6loCT/ZfyJ/EXnCDPT35FFgE=">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</latexit> (t) = (2 cos t, 2 sin t) 0  t  ⇡ 2

<latexit sha1_base64="td88SzNU4hpeTw/QK2DHzBeZkL0=">AAADEHicjVJNTxRBEH2MX4gfrHIiXiZuTDxtZlcSPRK5eISEBRKWkJmmFzvMzow9PSSbycb/wH/gilduxKv/wH8gN36Cr8vGqMRoT6bn9at6NVXVlVW5qV2SfJ2Lbt2+c/fe/P2FBw8fPV7sPHm6VZeNVXqoyry0O1la69wUeuiMy/VOZXU6yXK9nR2tefv2sba1KYtNN6303iQ9LMzYqNSR2u8sJ6Ncf4id7KOxTVU7qsysHcz2O92kl8iKb4J+AF2EtV52rjDCAUooNJhAo4AjzpGi5rOLPhJU5PbQkrNERuwaMyxQ29BL0yMle8T9kKfdwBY8+5i1qBX/kvO1VMZ4QU1JP0vs/xaLvZHInv1b7FZi+tym/GYh1oSsw3uy/9Jde/6vztfiMMYbqcGwpkoYX50KURrpis88/qUqxwgVOY8PaLfESpTXfY5FU0vtvrep2L+Jp2f9WQXfBpchS41jiTr9mX0rd2hor6SXUyLHXW6JI9H/cwBugq1Br/+qN9hY6a6+DcMxj2d4jpecgNdYxTusY8hsPuIUZ/gUnUTn0UX0+YdrNBc0S/htRV++A4lIpVg=</latexit>

(24)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

~r<latexit sha1_base64="N6i6loCT/ZfyJ/EXnCDPT35FFgE=">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</latexit> (t) = (2 cos t, 2 sin t) 0  t  ⇡ 2

<latexit sha1_base64="td88SzNU4hpeTw/QK2DHzBeZkL0=">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</latexit>

~r 0(t) = ( 2 sin t, 2 cos t)

(25)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

~r<latexit sha1_base64="N6i6loCT/ZfyJ/EXnCDPT35FFgE=">AAADE3icjVLBShxBEH2OJhpN4hqP8dC4BFYIy+wmoBdB9OJRIauCKzLTtqZxdmbo7llYFi/+g/+Qq169iVc/wD9Ibn5CqstWEiXEHqbn9at6NVXVlZaZti6Ob0ei0bFXr8cn3kxOvX33fro282HLFpWRqiOLrDA7aWJVpnPVcdplaqc0KumlmdpOj9e8fbuvjNVF/s0NSrXXS45yfahl4ojar811+0oK03ALy412VxZWuM+i3bU6F25hv1aPmzEv8Ry0AqgjrI2idocuDlBAokIPCjkc4QwJLD27aCFGSdwehsQZQprtCieYJG1FXoo8EmKPaT+i025gczr7mJbVkv6S0WtIKfCJNAX5GcL+b4LtFUf27L9iDzmmz21A3zTE6hHr8J3Y/+kePF+q87U4HGKJa9BUU8mMr06GKBV3xWcu/qjKUYSSOI8PyG4IS1Y+9FmwxnLtvrcJ23+yp2f9WQbfCr9Clgp9jjp4zH7Id6jJXnIvB4Qc7XxLNBKtpwPwHGy1m60vzfbm1/rKahiOCXzEPBo0AYtYwTo20KFsTvED57iIzqLL6Cq6vneNRoJmFn+t6OY37YKkpQ==</latexit> (t) = (2 cos t, 2 sin t) 0  t  ⇡ 2

<latexit sha1_base64="td88SzNU4hpeTw/QK2DHzBeZkL0=">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</latexit>

~r 0(t) = ( 2 sin t, 2 cos t)

<latexit sha1_base64="JTcCWgT+0032mC440KscGGnzOIU=">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</latexit>

k<latexit sha1_base64="0EAjtMEbUofZcQvgRz7KU8Cwew8=">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</latexit> ~r 0(t)k = 2

(26)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

~r<latexit sha1_base64="N6i6loCT/ZfyJ/EXnCDPT35FFgE=">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</latexit> (t) = (2 cos t, 2 sin t) 0  t  ⇡ 2

<latexit sha1_base64="td88SzNU4hpeTw/QK2DHzBeZkL0=">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</latexit>

~r 0(t) = ( 2 sin t, 2 cos t)

<latexit sha1_base64="JTcCWgT+0032mC440KscGGnzOIU=">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</latexit>

k<latexit sha1_base64="0EAjtMEbUofZcQvgRz7KU8Cwew8=">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</latexit> ~r 0(t)k = 2

Z b

a

f (~r(t))k~r 0(t)kdt

<latexit sha1_base64="3vmHk6eJU/vNM5qUrCOA76+lDKA=">AAADHXicjVJNb9NAEH01X6V8hXLkYhEB6SVyChIcq3LhWCTSVqpLZG82ZVXHttbrSFHKP+h/6H/gClduiCviH8CNn8Db6RYBFYK1vH77Zt54ZnbyujCNS5IvS9GFi5cuX1m+unLt+o2btzq3V7ebqrVKD1VVVHY3zxpdmFIPnXGF3q2tzqZ5oXfyw2fevjPTtjFV+dLNa70/zQ5KMzEqc6RGnQepKd0oe5VPeulMq9j23NpaenSK0/ghj+nR2I063aSfyIrPg0EAXYS1VXW+I8UYFRRaTKFRwhEXyNDw2cMACWpy+1iQs0RG7BpvsEJtSy9Nj4zsIfcDnvYCW/LsYzaiVvxLwddSGeM+NRX9LLH/Wyz2ViJ79m+xFxLT5zbnNw+xpmQdXpP9l+7M8391vhaHCZ5KDYY11cL46lSI0kpXfObxL1U5RqjJeTym3RIrUZ71ORZNI7X73mZi/yqenvVnFXxbfAtZaswk6vxn9gu5Q0N7Lb2cEznuckscicGfA3AebK/3B4/66y8edzc2w3As4y7uoccJeIINPMcWhszmGG/xDu+jk+hD9DH6dOoaLQXNHfy2os8/APr+qVY=</latexit>

(27)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

~r<latexit sha1_base64="N6i6loCT/ZfyJ/EXnCDPT35FFgE=">AAADE3icjVLBShxBEH2OJhpN4hqP8dC4BFYIy+wmoBdB9OJRIauCKzLTtqZxdmbo7llYFi/+g/+Qq169iVc/wD9Ibn5CqstWEiXEHqbn9at6NVXVlZaZti6Ob0ei0bFXr8cn3kxOvX33fro282HLFpWRqiOLrDA7aWJVpnPVcdplaqc0KumlmdpOj9e8fbuvjNVF/s0NSrXXS45yfahl4ojar811+0oK03ALy412VxZWuM+i3bU6F25hv1aPmzEv8Ry0AqgjrI2idocuDlBAokIPCjkc4QwJLD27aCFGSdwehsQZQprtCieYJG1FXoo8EmKPaT+i025gczr7mJbVkv6S0WtIKfCJNAX5GcL+b4LtFUf27L9iDzmmz21A3zTE6hHr8J3Y/+kePF+q87U4HGKJa9BUU8mMr06GKBV3xWcu/qjKUYSSOI8PyG4IS1Y+9FmwxnLtvrcJ23+yp2f9WQbfCr9Clgp9jjp4zH7Id6jJXnIvB4Qc7XxLNBKtpwPwHGy1m60vzfbm1/rKahiOCXzEPBo0AYtYwTo20KFsTvED57iIzqLL6Cq6vneNRoJmFn+t6OY37YKkpQ==</latexit> (t) = (2 cos t, 2 sin t) 0  t  ⇡ 2

<latexit sha1_base64="td88SzNU4hpeTw/QK2DHzBeZkL0=">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</latexit>

~r 0(t) = ( 2 sin t, 2 cos t)

<latexit sha1_base64="JTcCWgT+0032mC440KscGGnzOIU=">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</latexit>

k<latexit sha1_base64="0EAjtMEbUofZcQvgRz7KU8Cwew8=">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</latexit> ~r 0(t)k = 2

=

Z 2

0

(2 cos t)2 (2 sin t)22 dt

<latexit sha1_base64="O/n2Jph/19k6lzc74RVg3N5U/PU=">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</latexit>

Z b

a

f (~r(t))k~r 0(t)kdt

<latexit sha1_base64="3vmHk6eJU/vNM5qUrCOA76+lDKA=">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</latexit>

(28)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

~r<latexit sha1_base64="N6i6loCT/ZfyJ/EXnCDPT35FFgE=">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</latexit> (t) = (2 cos t, 2 sin t) 0  t  ⇡ 2

<latexit sha1_base64="td88SzNU4hpeTw/QK2DHzBeZkL0=">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</latexit>

~r 0(t) = ( 2 sin t, 2 cos t)

<latexit sha1_base64="JTcCWgT+0032mC440KscGGnzOIU=">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</latexit>

k<latexit sha1_base64="0EAjtMEbUofZcQvgRz7KU8Cwew8=">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</latexit> ~r 0(t)k = 2

=

Z 2

0

(2 cos t)2 (2 sin t)22 dt

<latexit sha1_base64="O/n2Jph/19k6lzc74RVg3N5U/PU=">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</latexit>

Z b

a

f (~r(t))k~r 0(t)kdt

<latexit sha1_base64="3vmHk6eJU/vNM5qUrCOA76+lDKA=">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</latexit>

(29)

Exemple

Z

C

x2 y2 ds

<latexit sha1_base64="RuI4yT3NQW5vD5nBgphgVkYVkN4=">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</latexit>

C

<latexit sha1_base64="3VMjrDSWrZOvhvAJzAxbjiPVxYI=">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</latexit>

Où est l’arc de cercle de rayon 2 du premier quadrant.

~r<latexit sha1_base64="N6i6loCT/ZfyJ/EXnCDPT35FFgE=">AAADE3icjVLBShxBEH2OJhpN4hqP8dC4BFYIy+wmoBdB9OJRIauCKzLTtqZxdmbo7llYFi/+g/+Qq169iVc/wD9Ibn5CqstWEiXEHqbn9at6NVXVlZaZti6Ob0ei0bFXr8cn3kxOvX33fro282HLFpWRqiOLrDA7aWJVpnPVcdplaqc0KumlmdpOj9e8fbuvjNVF/s0NSrXXS45yfahl4ojar811+0oK03ALy412VxZWuM+i3bU6F25hv1aPmzEv8Ry0AqgjrI2idocuDlBAokIPCjkc4QwJLD27aCFGSdwehsQZQprtCieYJG1FXoo8EmKPaT+i025gczr7mJbVkv6S0WtIKfCJNAX5GcL+b4LtFUf27L9iDzmmz21A3zTE6hHr8J3Y/+kePF+q87U4HGKJa9BUU8mMr06GKBV3xWcu/qjKUYSSOI8PyG4IS1Y+9FmwxnLtvrcJ23+yp2f9WQbfCr9Clgp9jjp4zH7Id6jJXnIvB4Qc7XxLNBKtpwPwHGy1m60vzfbm1/rKahiOCXzEPBo0AYtYwTo20KFsTvED57iIzqLL6Cq6vneNRoJmFn+t6OY37YKkpQ==</latexit> (t) = (2 cos t, 2 sin t) 0  t  ⇡ 2

<latexit sha1_base64="td88SzNU4hpeTw/QK2DHzBeZkL0=">AAADEHicjVJNTxRBEH2MX4gfrHIiXiZuTDxtZlcSPRK5eISEBRKWkJmmFzvMzow9PSSbycb/wH/gilduxKv/wH8gN36Cr8vGqMRoT6bn9at6NVXVlVW5qV2SfJ2Lbt2+c/fe/P2FBw8fPV7sPHm6VZeNVXqoyry0O1la69wUeuiMy/VOZXU6yXK9nR2tefv2sba1KYtNN6303iQ9LMzYqNSR2u8sJ6Ncf4id7KOxTVU7qsysHcz2O92kl8iKb4J+AF2EtV52rjDCAUooNJhAo4AjzpGi5rOLPhJU5PbQkrNERuwaMyxQ29BL0yMle8T9kKfdwBY8+5i1qBX/kvO1VMZ4QU1JP0vs/xaLvZHInv1b7FZi+tym/GYh1oSsw3uy/9Jde/6vztfiMMYbqcGwpkoYX50KURrpis88/qUqxwgVOY8PaLfESpTXfY5FU0vtvrep2L+Jp2f9WQXfBpchS41jiTr9mX0rd2hor6SXUyLHXW6JI9H/cwBugq1Br/+qN9hY6a6+DcMxj2d4jpecgNdYxTusY8hsPuIUZ/gUnUTn0UX0+YdrNBc0S/htRV++A4lIpVg=</latexit>

~r 0(t) = ( 2 sin t, 2 cos t)

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k<latexit sha1_base64="0EAjtMEbUofZcQvgRz7KU8Cwew8=">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</latexit> ~r 0(t)k = 2

=

Z 2

0

(2 cos t)2 (2 sin t)22 dt

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Z b

a

f (~r(t))k~r 0(t)kdt

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