Question

Calculate the air of the surface limited by the parabola which has the equation y=x(x) and the line passing through the points of abscissa -2 and 1 of the parabola.

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Answer to a math question Calculate the air of the surface limited by the parabola which has the equation y=x(x) and the line passing through the points of abscissa -2 and 1 of the parabola.

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Brice
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Pour trouver l'aire de la surface limitée par la parabole y=x^2 et la droite passant par les points d'abscisse -2 et 1 de la parabole, nous devons d'abord trouver les points d'intersection de la parabole et de la droite.

La droite passant par les points d'abscisse -2 et 1 de la parabole est définie par deux points (-2, (-2)^2) et (1, 1^2) . Donc, la droite a pour équation y = \frac{3}{3}x+4 .

Nous devons maintenant trouver les points d'intersection entre la parabole y=x^2 et la droite y = \frac{1}{3}x+4 :
x^2 = \frac{1}{3}x+4
x^2 - \frac{1}{3}x - 4 = 0

En résolvant cette équation quadratique, nous trouvons deux solutions pour x, à savoir x=-3 et x=4.

Pour calculer l'aire de la surface limitée par la parabole et la droite, nous devons trouver les limites d'intégration. Pour ce faire, nous trouvons les ordonnées des points d'intersection : (-3, (-3)^2) et (4, 4^2) .

L'aire recherchée est donc donnée par :
A = \int_{-3}^{4} (x^2 - \frac{1}{3}x - 4) \, dx
A = \left[\frac{x^3}{3} - \frac{x^2}{6} - 4x\right]_{-3}^{4}
A = (\frac{64}{3} - \frac{16}{6} - 16) - (-\frac{27}{3} + \frac{9}{6} + 12)
A = \frac{64}{3} - \frac{8}{3} - 16 + \frac{27}{3} - \frac{3}{2} - 12
A = \frac{64-8-48+27-6-72}{6} = \frac{27}{6} = \frac{9}{2}

\boxed{A = \frac{9}{2}}

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