15.6E: Ejercicios para la Sección 15.6

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Edwin “Jed” Herman & Gilbert Strang
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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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