Quiz — CoursIntegMultCurv.pdf

Question 1 sur 10 20:00

[{"id":76976,"question":"Quelle est la formule pour calculer l'aire entre deux courbes y = f(x) et y = g(x) sur [a, b] ?","option_a":"A = ∫[a,b] (f(x) + g(x)) dx","option_b":"B = ∫[a,b] |f(x) - g(x)| dx","option_c":"C = ∫[a,b] (f(x) * g(x)) dx","option_d":"D = ∫[a,b] (f(x) \/ g(x)) dx","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'aire entre deux courbes est donnée par l'intégrale de la valeur absolue de leur différence, car l'aire est toujours positive.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A = ∫[a,b] (f(x) + g(x)) dx\", \"b\": \"B = ∫[a,b] |f(x) - g(x)| ","_debug_options_count":4},{"id":76977,"question":"Si f(x) = x² et g(x) = x, quelle est l'aire entre ces deux courbes sur [0, 1] ?","option_a":"A = 1\/6","option_b":"B = 1\/3","option_c":"C = 1\/2","option_d":"D = 2\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'aire est ∫[0,1] (x - x²) dx = [x²\/2 - x³\/3] de 0 à 1 = (1\/2 - 1\/3) = 1\/6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A = 1\/6\", \"b\": \"B = 1\/3\", \"c\": \"C = 1\/2\", \"d\": \"D = 2\/3\"}}","_debug_options_count":4},{"id":76978,"question":"Vrai ou Faux : L'intégrale ∫[a,b] f(x) dx représente toujours une aire positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale peut être négative si f(x) est négative sur [a,b]. Pour obtenir une aire, on prend la valeur absolue.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":76979,"question":"Quelle méthode utiliser pour calculer l'aire entre deux courbes paramétrées x = f(t), y = g(t) et x = h(t), y = k(t) ?","option_a":"A = ∫ |f(t) - h(t)| dt","option_b":"B = ∫ |g(t) - k(t)| * √(f'(t)² + g'(t)²) dt","option_c":"C = ∫ (g(t) - k(t)) dt","option_d":"D = ∫ (f(t) - h(t)) * (g(t) - k(t)) dt","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour des courbes paramétrées, l'aire s'obtient en intégrant la différence des ordonnées multipliée par la norme du vecteur tangent.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A = ∫ |f(t) - h(t)| dt\", \"b\": \"B = ∫ |g(t) - k(t)| * √(f'(t","_debug_options_count":4},{"id":76980,"question":"Si deux courbes s'intersectent en x = 1 et x = 3, quelle est la borne inférieure de l'intégrale pour calculer l'aire entre elles ?","option_a":"A = 0","option_b":"B = 1","option_c":"C = 3","option_d":"D = -1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les bornes d'intégration sont les points d'intersection des courbes, ici x = 1 et x = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A = 0\", \"b\": \"B = 1\", \"c\": \"C = 3\", \"d\": \"D = -1\"}}","_debug_options_count":4},{"id":76981,"question":"Vrai ou Faux : L'aire entre deux courbes est toujours positive, même si l'une des fonctions est négative.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'aire est positive par définition, mais l'intégrale de la différence peut être négative si la fonction du bas est au-dessus de celle du haut.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":76982,"question":"Quelle est l'aire entre y = sin(x) et y = cos(x) sur [0, π\/4] ?","option_a":"A = √2 - 1","option_b":"B = 1 - √2\/2","option_c":"C = π\/4","option_d":"D = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'aire est ∫[0,π\/4] (cos(x) - sin(x)) dx = [sin(x) + cos(x)] de 0 à π\/4 = (√2\/2 + √2\/2) - (0 + 1) = √2 - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A = √2 - 1\", \"b\": \"B = 1 - √2\/2\", \"c\": \"C = π\/4\", \"d\": \"D = ","_debug_options_count":4},{"id":76983,"question":"Pour calculer l'aire entre deux courbes, faut-il toujours prendre la fonction du haut moins celle du bas ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, il faut prendre la différence absolue pour garantir un résultat positif, surtout si les courbes se croisent.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":76984,"question":"Quelle est l'aire entre y = e^x et y = x + 1 sur [0, 1] ?","option_a":"A = e - 2","option_b":"B = 1","option_c":"C = e + 1","option_d":"D = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'aire est ∫[0,1] (e^x - (x + 1)) dx = [e^x - x²\/2 - x] de 0 à 1 = (e - 1\/2 - 1) - (1 - 0 - 0) = e - 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A = e - 2\", \"b\": \"B = 1\", \"c\": \"C = e + 1\", \"d\": \"D = 0\"}}","_debug_options_count":4},{"id":76985,"question":"Vrai ou Faux : L'intégrale ∫[a,b] (f(x) - g(x)) dx donne directement l'aire entre deux courbes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, car si f(x) \u003C g(x) sur une partie de l'intervalle, l'intégrale sera négative. Il faut prendre la valeur absolue ou ajuster les bornes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]