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15.6E: Ejercicios para la Sección 15.6

  • Page ID
    116281
    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    En los ejercicios 1 - 12, la región\(R\) ocupada por una lámina se muestra en una gráfica. Encuentra la masa de\(R\) con la función de densidad\(\rho\).

    1. \(R\)es la región triangular con vértices\((0,0), \space (0,3)\), y\((6,0); \space \rho (x,y) = xy\).

    Un triángulo rectángulo delimitado por los ejes x e y y la línea y = x/2 + 3 negativo.

    Contestar
    \(\frac{27}{2}\)

    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. \(R\)es la región rectangular con vértices\((0,0), \space (0,3), \space (6,3) \) y\((6,0); \space \rho (x,y) = \sqrt{xy}\).

    Un rectángulo delimitado por los ejes x e y y las líneas x = 6 e y = 3.

    Contestar
    \(24\sqrt{2}\)

    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. \(R\)es la región trapezoidal determinada por las líneas\(y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2\), y\(x = 0; \space \rho (x,y) = 3xy\).

    Un trapecio delimitado por los ejes x e y, la línea y = 2, y la línea y = x/4 + 2.5 negativo.

    Contestar
    \(76\)

    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. \(R\)es el disco de radio\(2\) centrado en\((1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5\).

    Contestar
    \(8\pi\)

    8. \(R\)es el disco de la unidad;\(\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4\).

    Un círculo con radio 1 y centro del origen.

    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.

    Contestar
    \(\frac{\pi}{2}\)

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

    El cuarto de sección de una elipse en el primer cuadrante con centro el origen, eje mayor 2, y eje menor aproximadamente 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).

    Contestar
    \(2\)

    12. \(R\)es la región delimitada por\(y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1\), y\(y = 2; \space \rho (x,y) = 4(x + y)\).

    Una región compleja entre 2 y 1 que barre hacia abajo y hacia la derecha con límites y = 1/x e y = 2/x.

    En los ejercicios 13 - 24, considere una lámina ocupando la región\(R\) y teniendo la función de densidad\(\rho\) dada en el grupo anterior de ejercicios. Utilice un sistema de álgebra computacional (CAS) para responder a las siguientes preguntas.

    a. Encuentra los momentos\(M_x\) y\(M_y\) alrededor del\(x\) eje -eje y\(y\) -eje, respectivamente.

    b. Calcular y trazar el centro de masa de la lámina.

    c. [T] Utilice un CAS para ubicar el centro de masa en la gráfica 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]\(R\) es la región triangular con vértices\((0,0), \space (1,1)\), y\((0,5); \space \rho (x,y) = x + y\).

    15. [T]\(R\) es la región rectangular con vértices\((0,0), \space (0,3), \space (6,3)\), y\((6,0); \space \rho (x,y) = \sqrt{xy}\).

    Contestar

    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 rectángulo R delimitado por los ejes x e y y las líneas x = 6 e y = 3 con punto marcado (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]\(R\) es la región trapezoidal determinada por las líneas\(y = 0, \space y = 1, \space y = x,\) y\(y = -x + 3; \space \rho (x,y) = 2x + y\).

    19. [T]\(R\) es el disco de radio\(2\) centrado en\((1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5\).

    Contestar

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

    Un círculo con radio 2 centrado en (1, 2), que es tangente al eje x en (1, 0) y ha apuntado marcado en el centro (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 \(y = -x + 2; \space \rho (x,y) = 1\).

    Contestar

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

    Un cuadrado R con raíz cuadrada de longitud lateral de 2 girado 45 grados, con esquinas en el origen, (2, 0), (1, 1) y (negativo 1, 1). Un punto se marca en (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. Considera el sólido\(Q = \big\{ (x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\big\}\) con la función de densidad\(\rho(x,y,z) = x + y + 1\).

    a. encontrar la masa de\(Q\).

    b. Encontrar los momentos\(M_{xy}, \space M_{xz}\) y\(M_{yz}\) sobre el\(xy\) -plano,\(xz\) -plano, y\(yz\) -plano, respectivamente.

    c. Encuentra el centro de masa de\(Q\).

    43. [T] El sólido\(Q\) tiene la masa dada por la triple integral\(\displaystyle \int_{-1}^1 \int_0^{\pi/4} \int_0^1 r^2 \, dr \space d\theta \space dz.\)

    Utilice un CAS para responder a las siguientes preguntas.

    • Demostrar que el centro de masa de\(Q\) está ubicado en el\(xy\) -plano.
    • Grafica\(Q\) y localiza su centro de masa.
    Contestar

    \(\bar{x} = \frac{3\sqrt{2}}{2\pi}\),\(\bar{y} = \frac{3(2-\sqrt{2})}{2\pi}, \space \bar{z} = 0\); 2. el sólido\(Q\) y su centro de masa se muestran en la siguiente figura.

    Una cuña de un cilindro en el primer cuadrante con altura 2, radio 1 y ángulo aproximadamente 45 grados. Un punto se marca en (3 veces la raíz cuadrada de 2/ (2 pi), 3 veces (2 menos la raíz cuadrada de 2)/(2 pi), 0).

    44. El sólido

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