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15.6E : Exercices pour la section 15.6

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    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Dans les exercices 1 à 12, la région\(R\) occupée par une lame est représentée sur un graphique. Déterminez la masse de\(R\) avec la fonction de densité\(\rho\).

    1. \(R\)est la région triangulaire avec des sommets\((0,0), \space (0,3)\), et\((6,0); \space \rho (x,y) = xy\).

    Un triangle droit délimité par les axes x et y et par la ligne y = négatif x/2 + 3.

    Réponse
    \(\frac{27}{2}\)

    2. \(R\) is the triangular region with vertices \((0,0), \space (1,1)\), and \((0,5); \space \rho (x,y) = x + y\).

    A triangle bounded by the y axis, the line x = y, and the line y = negative 4x + 5.

    3. \(R\)est la région rectangulaire avec des sommets\((0,0), \space (0,3), \space (6,3) \) et\((6,0); \space \rho (x,y) = \sqrt{xy}\).

    Un rectangle délimité par les axes x et y et par les lignes x = 6 et y = 3.

    Réponse
    \(24\sqrt{2}\)

    4. \(R\) is the rectangular region with vertices \((0,1), \space (0,3), \space (3,3)\) and \( (3,1); \space \rho (x,y) = x^2y\).

    A rectangle bounded by the y axis, the lines y = 1 and 3, and the line x = 3.

    5. \(R\)est la région trapézoïdale déterminée par les droites\(y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2\), et\(x = 0; \space \rho (x,y) = 3xy\).

    Un trapèze délimité par les axes x et y, la droite y = 2 et la droite y = négatif x/4 + 2,5.

    Réponse
    \(76\)

    6. \(R\) is the trapezoidal region determined by the lines \(y = 0, \space y = 1, \space y = x\) and \(y = -x + 3; \space \rho (x,y) = 2x + y\).

    A trapezoid bounded by the x axis, the line y = 1, the line y = x, and the line y = negative x + 3.

    7. \(R\)est le disque de rayon\(2\) centré sur\((1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5\).

    Réponse
    \(8\pi\)

    8. \(R\)est le disque unitaire ;\(\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4\).

    Un cercle dont le rayon est de 1 et dont l'origine est centrée.

    9. \(R\) is the region enclosed by the ellipse \(x^2 + 4y^2 = 1; \space \rho(x,y) = 1\).

    An ellipse with center the origin, major axis 2, and minor axis 0.5.

    Réponse
    \(\frac{\pi}{2}\)

    10. \(R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}\).

    Le quart de section d'une ellipse dans le premier quadrant avec pour centre l'origine, le grand axe 2 et le petit axe d'environ 0,64.

    11. \(R\) is the region bounded by \(y = x, \space y = -x, \space y = x + 2, \space y = -x + 2; \space \rho(x,y) = 1\).

    A square with side length square root of 2 rotated 45 degrees, with corners at the origin, (2, 0), (1, 1), and (negative 1, 1).

    Réponse
    \(2\)

    12. \(R\)est la région délimitée par\(y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1\), et\(y = 2; \space \rho (x,y) = 4(x + y)\).

    Région complexe comprise entre 2 et 1 qui défile vers le bas et vers la droite avec des limites y = 1/x et y = 2/x.

    Dans les exercices 13 à 24, considérez une lame occupant la région\(R\) et ayant la fonction de densité\(\rho\) donnée dans le groupe d'exercices précédent. Utilisez un système d'algèbre informatique (CAS) pour répondre aux questions suivantes.

    a. Trouvez les moments\(M_x\) et\(M_y\) autour de l'\(x\)axe -et de\(y\) l'axe -, respectivement.

    b. Calculez et tracez le centre de masse de la lame.

    c. [T] Utilisez un CAS pour localiser le centre de gravité sur le graphique de\(R\).

    13. [T]\(R\) is the triangular region with vertices \((0,0), \space (0,3)\), and \((6,0); \space \rho (x,y) = xy\).

    Answer

    a. \(M_x = \frac{81}{5}, \space M_y = \frac{162}{5}\);
    b. \(\bar{x} = \frac{12}{5}, \space \bar{y} = \frac{6}{5}\);
    c.

    A triangular region R bounded by the x and y axes and the line y = negative x/2 + 3, with a point marked at (12/5, 6/5).

    14. [T]\(R\) est la région triangulaire avec des sommets\((0,0), \space (1,1)\), et\((0,5); \space \rho (x,y) = x + y\).

    15. [T]\(R\) est la région rectangulaire avec des sommets\((0,0), \space (0,3), \space (6,3)\), et\((6,0); \space \rho (x,y) = \sqrt{xy}\).

    Réponse

    a.\(M_x = \frac{216\sqrt{2}}{5}, \space M_y = \frac{432\sqrt{2}}{5}\) ;
    b.\(\bar{x} = \frac{18}{5}, \space \bar{y} = \frac{9}{5}\) ;
    c.

    Un rectangle R délimité par les axes x et y et par les lignes x = 6 et y = 3 avec un point marqué (18/5, 9/5).

    16. [T]\(R\) is the rectangular region with vertices \((0,1), \space (0,3), \space (3,3)\), and \((3,1); \space \rho (x,y) = x^2y\).

    17. [T] \(R\) is the trapezoidal region determined by the lines \(y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2\), and \(x = 0; \space \rho (x,y) = 3xy\).

    Answer

    a. \(M_x = \frac{368}{5}, \space M_y = \frac{1552}{5}\);
    b. \(\bar{x} = \frac{92}{95}, \space \bar{y} = \frac{388}{95}\);
    c.

    A trapezoid R bounded by the x and y axes, the line y = 2, and the line y = negative x/4 + 2.5 with the point marked (92/95, 388/95).

    18. [T]\(R\) est la région trapézoïdale déterminée par les lignes\(y = 0, \space y = 1, \space y = x,\) et\(y = -x + 3; \space \rho (x,y) = 2x + y\).

    19. [T]\(R\) est le disque dont le rayon est\(2\) centré sur\((1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5\).

    Réponse

    a.\(M_x = 16\pi, \space M_y = 8\pi\) ;
    b.\(\bar{x} = 1, \space \bar{y} = 2\) ;
    c.

    Cercle de rayon 2 centré sur (1, 2), tangent à l'axe x en (1, 0) et dont la pointe est marquée au centre (1, 2).

    20. [T]\(R\) is the unit disk; \(\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4\).

    21. [T] \(R\) is the region enclosed by the ellipse \(x^2 + 4y^2 = 1; \space \rho(x,y) = 1\).

    Answer

    a. \(M_x = 0, \space M_y = 0)\);
    b. \(\bar{x} = 0, \space \bar{y} = 0\);
    c.

    An ellipse R with center the origin, major axis 2, and minor axis 0.5, with point marked at the origin.

    22. [T]\(R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}\).

    23. [T]\(R\) is the region bounded by \(y = x, \space y = -x, \space y = x + 2\), and \(y = -x + 2; \space \rho (x,y) = 1\).

    Réponse

    a.\(M_x = 2, \space M_y = 0)\) ;
    b.\(\bar{x} = 0, \space \bar{y} = 1\) ;
    c.

    Un carré R avec une racine carrée de 2 pivotant de 45 degrés, avec des coins à l'origine, (2, 0), (1, 1) et (négatif 1, 1). Un point est marqué à (0, 1).

    24. [T]\(R\) is the region bounded by \(y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1\), and \(y = 2; \space \rho (x,y) = 4(x + y)\).

    In exercises 25 - 36, consider a lamina occupying the region \(R\) and having the density function \(\rho\) given in the first two groups of Exercises.

    a. Find the moments of inertia \(I_x, \space I_y\) and \(I_0\) about the \(x\)-axis, \(y\)-axis, and origin, respectively.

    b. Find the radii of gyration with respect to the \(x\)-axis, \(y\)-axis, and origin, respectively.

    25. \(R\) is the triangular region with vertices \((0,0), \space (0,3)\), and \((6,0); \space \rho (x,y) = xy\).

    Answer
    a. \(I_x = \frac{243}{10}, \space I_y = \frac{486}{5}\), and \(I_0 = \frac{243}{2}\);
    b. \(R_x = \frac{3\sqrt{5}}{5}, \space R_y = \frac{6\sqrt{5}}{5}\), and \(R_0 = 3\)

    26. \(R\) is the triangular region with vertices \((0,0), \space (1,1)\), and \((0,5); \space \rho (x,y) = x + y\).

    27. \(R\) is the rectangular region with vertices \((0,0), \space (0,3), \space (6,3)\), and \((6,0); \space \rho (x,y) = \sqrt{xy}\).

    Answer
    a. \(I_x = \frac{2592\sqrt{2}}{7}, \space I_y = \frac{648\sqrt{2}}{7}\), and \(I_0 = \frac{3240\sqrt{2}}{7}\);
    b. \(R_x = \frac{6\sqrt{21}}{7}, \space R_y = \frac{3\sqrt{21}}{7}\), and \(R_0 = \frac{3\sqrt{106}}{7}\)

    28. \(R\) is the rectangular region with vertices \((0,1), \space (0,3), \space (3,3)\), and \((3,1); \space \rho (x,y) = x^2y\).

    29. \(R\) is the trapezoidal region determined by the lines \(y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2\), and x = 0; \space \rho (x,y) = 3xy\).

    Answer
    a. \(I_x = 88, \space I_y = 1560\), and \(I_0 = 1648\);
    b. \(R_x = \frac{\sqrt{418}}{19}, \space R_y = \frac{\sqrt{7410}}{10}\), and \(R_0 = \frac{2\sqrt{1957}}{19}\)

    30. \(R\) is the trapezoidal region determined by the lines \(y = 0, \space y = 1, \space y = x\), and y = -x + 3; \space \rho (x,y) = 2x + y\).

    31. \(R\) is the disk of radius \(2\) centered at \((1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5\).

    Answer
    a. \(I_x = \frac{128\pi}{3}, \space I_y = \frac{56\pi}{3}\), and \(I_0 = \frac{184\pi}{3}\);
    b. \(R_x = \frac{4\sqrt{3}}{3}, \space R_y = \frac{\sqrt{21}}{2}\), and \(R_0 = \frac{\sqrt{69}}{3}\)

    32. \(R\) is the unit disk; \(\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4\).

    33. \(R\) is the region enclosed by the ellipse \(x^2 + 4y^2 = 1; \space \rho(x,y) = 1\).

    Answer
    a. \(I_x = \frac{\pi}{32}, \space I_y = \frac{\pi}{8}\), and \(I_0 = \frac{5\pi}{32}\);
    b. \(R_x = \frac{1}{4}, \space R_y = \frac{1}{2}\), and \(R_0 = \frac{\sqrt{5}}{4}\)

    34. \(R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}\).

    35. \(R\) is the region bounded by \(y = x, \space y = -x, \space y = x + 2\), and \(y = -x + 2; \space \rho (x,y) = 1\).

    Answer
    a. \(I_x = \frac{7}{3}, \space I_y = \frac{1}{3}\), and \(I_0 = \frac{8}{3}\);
    b. \(R_x = \frac{\sqrt{42}}{6}, \space R_y = \frac{\sqrt{6}}{6}\), and \(R_0 = \frac{2\sqrt{3}}{3}\)

    36. \(R\) is the region bounded by \(y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1\), and \(y = 2; \space \rho (x,y) = 4(x + y)\).

    37. Let \(Q\) be the solid unit cube. Find the mass of the solid if its density \(\rho\) is equal to the square of the distance of an arbitrary point of \(Q\) to the \(xy\)-plane.

    Answer
    \(m = \frac{1}{3}\)

    38. Let \(Q\) be the solid unit hemisphere. Find the mass of the solid if its density \(\rho\) is proportional to the distance of an arbitrary point of \(Q\) to the origin.

    39. The solid \(Q\) of constant density \(1\) is situated inside the sphere \(x^2 + y^2 + z^2 = 16\) and outside the sphere \(x^2 + y^2 + z^2 = 1\). Show that the center of mass of the solid is not located within the solid.

    40. Find the mass of the solid \(Q = \big\{ (x,y,z) \,|\, 1 \leq x^2 + z^2 \leq 25, \space y \leq 1 - x^2 - z^2 \big\}\) whose density is \(\rho (x,y,z) = k\), where \(k > 0\).

    41. [T] The solid \(Q = \big\{ (x,y,z) \,|\, x^2 + y^2 \leq 9, \space 0 \leq z \leq 1, \space x \geq 0, \space y \geq 0\big\}\) has density equal to the distance to the \(xy\)-plane. Use a CAS to answer the following questions.

    a. Find the mass of \(Q\).

    b. Find the moments \(M_{xy}, \space M_{xz}\) and \(M_{yz}\) about the \(xy\)-plane, \(xz\)-plane, and \(yz\)-plane, respectively.

    c. Find the center of mass of \(Q\).

    d. Graph \(Q\) and locate its center of mass.

    Answer

    a. \(m = \frac{9\pi}{4}\);
    b. \(M_{xy} = \frac{3\pi}{2}, \space M_{xz} = \frac{81}{8}, \space M_{yz} = \frac{81}{8}\);
    c. \(\bar{x} = \frac{9}{2\pi}, \space \bar{y} = \frac{9}{2\pi}, \space \bar{z} = \frac{2}{3}\);
    d.

    A quarter cylinder in the first quadrant with height 1 and radius 3. A point is marked at (9/(2 pi), 9/(2 pi), 2/3).

    42. Considérez le solide\(Q = \big\{ (x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\big\}\) avec la fonction de densité\(\rho(x,y,z) = x + y + 1\).

    a. Déterminez la masse de\(Q\).

    b. Trouvez les moments\(M_{xy}, \space M_{xz}\) et les\(M_{yz}\) informations relatives au\(xy\) plan, au\(xz\) plan et au\(yz\) plan, respectivement.

    c. Déterminez le centre de gravité de\(Q\).

    43. [T] Le solide\(Q\) a la masse donnée par la triple intégrale\(\displaystyle \int_{-1}^1 \int_0^{\pi/4} \int_0^1 r^2 \, dr \space d\theta \space dz.\)

    Utilisez un CAS pour répondre aux questions suivantes.

    • Montrez que le centre de gravité de\(Q\) est situé dans le\(xy\) plan.
    • Tracez\(Q\) et localisez son centre de gravité.
    Réponse

    \(\bar{x} = \frac{3\sqrt{2}}{2\pi}\),\(\bar{y} = \frac{3(2-\sqrt{2})}{2\pi}, \space \bar{z} = 0\) ; 2. le solide\(Q\) et son centre de masse sont représentés sur la figure suivante.

    Un coin formé par un cylindre dans le premier quadrant avec une hauteur de 2, un rayon de 1 et un angle d'environ 45 degrés. Un point est marqué à (3 fois la racine carrée de 2/ (2 pi), 3 fois (2 moins la racine carrée de 2)/(2 pi), 0).

    44. Le solide

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