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1.4 LONGUEUR D’ARC ET ABSCISSE CURVILIGNE

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Texte intégral

(1)

cours 4

1.4 LONGUEUR

D’ARC ET ABSCISSE

CURVILIGNE

(2)

Au dernier cours, nous avons vu

(3)

Au dernier cours, nous avons vu

Dérivées de fonctions vectorielles.

(4)

Au dernier cours, nous avons vu

Dérivées de fonctions vectorielles.

Règles de dérivation.

(5)

Au dernier cours, nous avons vu

Dérivées de fonctions vectorielles.

Règles de dérivation.

Vecteur tangent unitaire.

(6)

Aujourd’hui, nous allons voir

(7)

Aujourd’hui, nous allons voir

Intégrale d’une fonction vectorielle

(8)

Aujourd’hui, nous allons voir

Intégrale d’une fonction vectorielle La longueur d’une courbe

(9)

Aujourd’hui, nous allons voir

Intégrale d’une fonction vectorielle La longueur d’une courbe

L’abscisse curviligne

(10)

Aujourd’hui, nous allons voir

Intégrale d’une fonction vectorielle La longueur d’une courbe

L’abscisse curviligne Repère de Frenet

(11)

L’équation vectorielle d’une droite dans estR<latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">AAACznicjVHLSsNAFD2Nr1pfVZdugkVwVRIVdFl047KKfUBbZZJO26F5kUwKpRS3/oBb/SzxD/QvvDOmoBbRCUnOnHvOnbn3OpEnEmlZrzljYXFpeSW/Wlhb39jcKm7v1JMwjV1ec0MvjJsOS7gnAl6TQnq8GcWc+Y7HG87wQsUbIx4nIgxu5DjiHZ/1A9ETLpNEtdo+kwPHMa9vj++KJats6WXOAzsDJWSrGhZf0EYXIVyk8MERQBL2wJDQ04INCxFxHUyIiwkJHeeYokDelFScFIzYIX37tGtlbEB7lTPRbpdO8eiNyWnigDwh6WLC6jRTx1OdWbG/5Z7onOpuY/o7WS6fWIkBsX/5Zsr/+lQtEj2c6RoE1RRpRlXnZllS3RV1c/NLVZIyRMQp3KV4TNjVzlmfTe1JdO2qt0zH37RSsWrvZtoU7+qWNGD75zjnQf2obBO+OilVzrNR57GHfRzSPE9RwSWqqOmOP+IJz0bVGBlT4/5TauQyzy6+LePhA+REkzY=</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit> 3

(12)

L’équation vectorielle d’une droite dans estR<latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit> 3

OP<latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">AAAC3nicjVHLSsNAFD2Nr1pfUVfiJlgEQSipCLoRqm7cWcE+oC01mU41NE1CMimUUty5E7f+gFv9HPEP9C+8M01BLaITkpw5954zc++1A9eJhGm+pbSp6ZnZufR8ZmFxaXlFX10rR34cMl5ivuuHVduKuOt4vCQc4fJqEHKra7u8YndOZbzS42Hk+N6l6Ae80bWuPaftMEsQ1dQ36j3OBufF4dEIHA93OxIZraaeNXOmWsYkyCcgi2QVff0VdbTggyFGFxweBGEXFiJ6asjDREBcAwPiQkKOinMMkSFtTFmcMixiO/S9pl0tYT3aS89IqRmd4tIbktLANml8ygsJy9MMFY+Vs2R/8x4oT3m3Pv3txKtLrMANsX/pxpn/1claBNo4VDU4VFOgGFkdS1xi1RV5c+NLVYIcAuIkblE8JMyUctxnQ2kiVbvsraXi7ypTsnLPktwYH/KWNOD8z3FOgvJeLk/4Yj9bOElGncYmtrBD8zxAAWcookTet3jCM160K+1Ou9ceRqlaKtGs49vSHj8B456Zag==</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit> ~ = OA~ + kd~

(13)

Donc l’équation de la droite tangente à la courbe est L’équation vectorielle d’une droite dans estR<latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit> 3

OP<latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">AAAC3nicjVHLSsNAFD2Nr1pfUVfiJlgEQSipCLoRqm7cWcE+oC01mU41NE1CMimUUty5E7f+gFv9HPEP9C+8M01BLaITkpw5954zc++1A9eJhGm+pbSp6ZnZufR8ZmFxaXlFX10rR34cMl5ivuuHVduKuOt4vCQc4fJqEHKra7u8YndOZbzS42Hk+N6l6Ae80bWuPaftMEsQ1dQ36j3OBufF4dEIHA93OxIZraaeNXOmWsYkyCcgi2QVff0VdbTggyFGFxweBGEXFiJ6asjDREBcAwPiQkKOinMMkSFtTFmcMixiO/S9pl0tYT3aS89IqRmd4tIbktLANml8ygsJy9MMFY+Vs2R/8x4oT3m3Pv3txKtLrMANsX/pxpn/1claBNo4VDU4VFOgGFkdS1xi1RV5c+NLVYIcAuIkblE8JMyUctxnQ2kiVbvsraXi7ypTsnLPktwYH/KWNOD8z3FOgvJeLk/4Yj9bOElGncYmtrBD8zxAAWcookTet3jCM160K+1Ou9ceRqlaKtGs49vSHj8B456Zag==</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit> ~ = OA~ + kd~

(14)

Donc l’équation de la droite tangente à la courbe est L’équation vectorielle d’une droite dans estR<latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit> 3

OP<latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit> ~ = OA~ + kd~

OP<latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit> ~ = ~r(a) + k~r 0(a)

(15)

Donc l’équation de la droite tangente à la courbe est L’équation vectorielle d’une droite dans estR<latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">AAACznicjVHLSsNAFD2Nr1pfVZdugkVwVRIVdFl047KKfUBbZZJO26F5kUwKpRS3/oBb/SzxD/QvvDOmoBbRCUnOnHvOnbn3OpEnEmlZrzljYXFpeSW/Wlhb39jcKm7v1JMwjV1ec0MvjJsOS7gnAl6TQnq8GcWc+Y7HG87wQsUbIx4nIgxu5DjiHZ/1A9ETLpNEtdo+kwPHMa9vj++KJats6WXOAzsDJWSrGhZf0EYXIVyk8MERQBL2wJDQ04INCxFxHUyIiwkJHeeYokDelFScFIzYIX37tGtlbEB7lTPRbpdO8eiNyWnigDwh6WLC6jRTx1OdWbG/5Z7onOpuY/o7WS6fWIkBsX/5Zsr/+lQtEj2c6RoE1RRpRlXnZllS3RV1c/NLVZIyRMQp3KV4TNjVzlmfTe1JdO2qt0zH37RSsWrvZtoU7+qWNGD75zjnQf2obBO+OilVzrNR57GHfRzSPE9RwSWqqOmOP+IJz0bVGBlT4/5TauQyzy6+LePhA+REkzY=</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit> 3

OP<latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit> ~ = OA~ + kd~

OP<latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit> ~ = ~r(a) + k~r 0(a)

(16)

Donc l’équation de la droite tangente à la courbe est L’équation vectorielle d’une droite dans estR<latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">AAACznicjVHLSsNAFD2Nr1pfVZdugkVwVRIVdFl047KKfUBbZZJO26F5kUwKpRS3/oBb/SzxD/QvvDOmoBbRCUnOnHvOnbn3OpEnEmlZrzljYXFpeSW/Wlhb39jcKm7v1JMwjV1ec0MvjJsOS7gnAl6TQnq8GcWc+Y7HG87wQsUbIx4nIgxu5DjiHZ/1A9ETLpNEtdo+kwPHMa9vj++KJats6WXOAzsDJWSrGhZf0EYXIVyk8MERQBL2wJDQ04INCxFxHUyIiwkJHeeYokDelFScFIzYIX37tGtlbEB7lTPRbpdO8eiNyWnigDwh6WLC6jRTx1OdWbG/5Z7onOpuY/o7WS6fWIkBsX/5Zsr/+lQtEj2c6RoE1RRpRlXnZllS3RV1c/NLVZIyRMQp3KV4TNjVzlmfTe1JdO2qt0zH37RSsWrvZtoU7+qWNGD75zjnQf2obBO+OilVzrNR57GHfRzSPE9RwSWqqOmOP+IJz0bVGBlT4/5TauQyzy6+LePhA+REkzY=</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit><latexit sha1_base64="Doqolex/8a6hKe+63cyWJs+DIos=">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</latexit> 3

OP<latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">AAAC3nicjVHLSsNAFD2Nr1pfUVfiJlgEQSipCLoRqm7cWcE+oC01mU41NE1CMimUUty5E7f+gFv9HPEP9C+8M01BLaITkpw5954zc++1A9eJhGm+pbSp6ZnZufR8ZmFxaXlFX10rR34cMl5ivuuHVduKuOt4vCQc4fJqEHKra7u8YndOZbzS42Hk+N6l6Ae80bWuPaftMEsQ1dQ36j3OBufF4dEIHA93OxIZraaeNXOmWsYkyCcgi2QVff0VdbTggyFGFxweBGEXFiJ6asjDREBcAwPiQkKOinMMkSFtTFmcMixiO/S9pl0tYT3aS89IqRmd4tIbktLANml8ygsJy9MMFY+Vs2R/8x4oT3m3Pv3txKtLrMANsX/pxpn/1claBNo4VDU4VFOgGFkdS1xi1RV5c+NLVYIcAuIkblE8JMyUctxnQ2kiVbvsraXi7ypTsnLPktwYH/KWNOD8z3FOgvJeLk/4Yj9bOElGncYmtrBD8zxAAWcookTet3jCM160K+1Ou9ceRqlaKtGs49vSHj8B456Zag==</latexit><latexit sha1_base64="H7S79+DVJ4RsGB/Xhn3Nm3zQpWE=">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</latexit> ~ = OA~ + kd~

OP<latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">AAAC5XicjVHLSsNAFD3GV31XXboJFrEiSCqCboSiG3dWsFawUibTsYamSZhMClLcunMnbv0Bt/or4h/oX3hnnIJaRCckOffce87MnesnYZAqz3sdcoZHRsfGcxOTU9Mzs3P5+YWTNM4kF1Ueh7E89VkqwiASVRWoUJwmUrCOH4qa397X+VpXyDSIo2N1lYjzDmtFwUXAmSKqkXfrXcF7h5XrXQ1cWWRr6+1PWHdXKWrkC96GZ5Y7CEoWFGBXJc6/oI4mYnBk6EAggiIcgiGl5wwleEiIO0ePOEkoMHmBa0ySNqMqQRWM2DZ9WxSdWTaiWHumRs1pl5BeSUoXK6SJqU4S1ru5Jp8ZZ83+5t0znvpsV/T3rVeHWIVLYv/S9Sv/q9O9KFxgx/QQUE+JYXR33Lpk5lb0yd0vXSlySIjTuEl5SZgbZf+eXaNJTe/6bpnJv5lKzeqY29oM7/qUNODSz3EOgpPNjRLho61Cec+OOoclLKNI89xGGQeooEreN3jEE56dlnPr3Dn3n6XOkNUs4ttyHj4A1PKazw==</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">AAAC5XicjVHLSsNAFD3GV31XXboJFrEiSCqCboSiG3dWsFawUibTsYamSZhMClLcunMnbv0Bt/or4h/oX3hnnIJaRCckOffce87MnesnYZAqz3sdcoZHRsfGcxOTU9Mzs3P5+YWTNM4kF1Ueh7E89VkqwiASVRWoUJwmUrCOH4qa397X+VpXyDSIo2N1lYjzDmtFwUXAmSKqkXfrXcF7h5XrXQ1cWWRr6+1PWHdXKWrkC96GZ5Y7CEoWFGBXJc6/oI4mYnBk6EAggiIcgiGl5wwleEiIO0ePOEkoMHmBa0ySNqMqQRWM2DZ9WxSdWTaiWHumRs1pl5BeSUoXK6SJqU4S1ru5Jp8ZZ83+5t0znvpsV/T3rVeHWIVLYv/S9Sv/q9O9KFxgx/QQUE+JYXR33Lpk5lb0yd0vXSlySIjTuEl5SZgbZf+eXaNJTe/6bpnJv5lKzeqY29oM7/qUNODSz3EOgpPNjRLho61Cec+OOoclLKNI89xGGQeooEreN3jEE56dlnPr3Dn3n6XOkNUs4ttyHj4A1PKazw==</latexit><latexit sha1_base64="2Awddmdbya/AE8AnkIfNJhpxk24=">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</latexit> ~ = ~r(a) + k~r 0(a)

(17)

Exemple

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

<latexit sha1_base64="jwisrcU9OPLyYxt5s5F2mO2Uf7M=">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</latexit><latexit sha1_base64="jwisrcU9OPLyYxt5s5F2mO2Uf7M=">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</latexit><latexit sha1_base64="jwisrcU9OPLyYxt5s5F2mO2Uf7M=">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</latexit><latexit sha1_base64="jwisrcU9OPLyYxt5s5F2mO2Uf7M=">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</latexit>

Hélice

(18)

Exemple

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

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Hélice

(19)

Exemple

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

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Hélice

(20)

Exemple

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

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Hélice

~r 0(t) = ( sin t, cos t, 1)

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(21)

Exemple

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

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Hélice

~r 0(t) = ( sin t, cos t, 1)

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L’équation de la droite tangente en t = 2

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(22)

Exemple

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

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Hélice

~r 0(t) = ( sin t, cos t, 1)

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L’équation de la droite tangente en t = 2

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(x, y, z) = ~r ⇣ ⇡ 2

⌘ + k~r 0 ⇣ ⇡ 2

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