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15.4E : Exercices pour la section 15.4

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    197460

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Dans les exercices 1 à 4, évaluez les intégrales triples sur la boîte pleine rectangulaire\(B\).

    1. \(\displaystyle \iiint_B (2x + 3y^2 + 4z^3) \, dV,\)où\(B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq 2, \, 0 \leq z \leq 3\big\}\)

    Réponse
    \(192\)

    2. \(\displaystyle \iiint_B (xy + yz + xz) \, dV,\)où\(B = \big\{(x,y,z) \,|\, 1 \leq x \leq 2, \, 0 \leq y \leq 2, \, 1 \leq z \leq 3\big\}\)

    3. \(\displaystyle \iiint_B (x \cos y + z) \, dV,\)où\(B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq \pi, \, -1 \leq z \leq 1\big\}\)

    Réponse
    \(0\)

    4. \(\displaystyle \iiint_B (z \sin x + y^2) \, dV,\)où\(B = \big\{(x,y,z) \,|\, 0 \leq x \leq \pi, \, 0 \leq y \leq 1, \, -1 \leq z \leq 2\big\}\)

    Dans les exercices 5 à 8, modifiez l'ordre d'intégration en intégrant d'abord par rapport à\(z\)\(x\), puis ensuite\(y\).

    5. \(\displaystyle \int_0^1 \int_1^2 \int_2^3 (x^2 + \ln y + z) \, dx \, dy \, dz\)

    Réponse
    \(\displaystyle \int_0^1 \int_1^2 \int_2^3 (x^2 + \ln y + z) \, dx \, dy \, dz = \frac{35}{6} + 2 \ln 2\)

    6. \(\displaystyle \int_0^1 \int_{-1}^1 \int_0^3 (ze^x + 2y) \, dx \, dy \, dz\)

    7. \(\displaystyle \int_{-1}^2 \int_1^3 \int_0^4 \left(x^2z + \frac{1}{y}\right) \, dx \, dy \, dz\)

    Réponse
    \(\displaystyle \int_{-1}^2 \int_1^3 \int_0^4 \left(x^2z + \frac{1}{y}\right) \, dx \, dy \, dz = 64 + 12 \ln 3\)

    8. \(\displaystyle \int_1^2 \int_{-2}^{-1} \int_0^1 \frac{x + y}{z} \, dx \, dy \, dz\)

    9. Let\(F\)\(G\), et\(H\) être des fonctions continues sur\([a,b]\)\([c,d]\), et\([e,f]\), respectivement, où\(a, \, b, \, c, \, d, \, e\), et\(f\) sont des nombres réels tels que\(a < b, \, c < d\), et\(e < f\). Montrez que

    \[\int_a^b \int_c^d \int_e^f F (x) \, G (y) \, H(z) \, dz \, dy \, dx = \left(\int_a^b F(x) \, dx \right) \left(\int_c^d G(y) \, dy \right) \left(\int_e^f H(z) \, dz \right).\nonumber \]

    10. Let\(F\)\(G\), et\(H\) be des fonctions différentielles sur\([a,b]\)\([c,d]\), et\([e,f]\), respectivement, où\(a, \, b, \, c, \, d, \, e\), et\(f\) sont des nombres réels tels que\(a < b, \, c < d\), et\(e < f\). Montrez que

    \[\int_a^b \int_c^d \int_e^f F' (x) \, G' (y) \, H'(z) \, dz \, dy \, dx = [F (b) - F (a)] \, [G(d) - G(c)] \, H(f) - H(e)].\nonumber \]

    Dans les exercices 11 à 14, évaluez les intégrales triples sur la région délimitée

    \(E = \big\{(x,y,z) \,|\, a \leq x \leq b, \, h_1 (x) \leq y \leq h_2 (x), \, e \leq z \leq f \big\}.\)

    11. \(\displaystyle \iiint_E (2x + 5y + 7z) \, dV, \)où\(E = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq -x + 1, \, 1 \leq z \leq 2\big\}\)

    Réponse
    \(\frac{77}{12}\)

    12. \(\displaystyle \iiint_E (y \ln x + z) \, dV,\)où\(E = \big\{(x,y,z) \,|\, 1 \leq x \leq e, \, 0 \leq y \ln x, \, 0 \leq z \leq 1\big\}\)

    13. \(\displaystyle \iiint_E (\sin x + \sin y) \, dV,\)où\(E = \big\{(x,y,z) \,|\, 0 \leq x \leq \frac{\pi}{2}, \, -\cos x \leq y \cos x, \, -1 \leq z \leq 1 \big\}\)

    Réponse
    \(2\)

    14. \(\displaystyle \iiint_E (xy + yz + xz ) dV\)où\(E = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, -x^2 \leq y \leq x^2, \, 0 \leq z \leq 1 \big\}\)

    Dans les exercices 15 à 18, évaluez les intégrales triples sur la région délimitée indiquée\(E\).

    15. \(\displaystyle \iiint_E (x + 2yz) \, dV,\)où\(E = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq x, \, 0 \leq z \leq 5 - x - y \big\}\)

    Réponse
    \(\frac{430}{120}\)

    16. \(\displaystyle \iiint_E (x^3 + y^3 + z^3) \, dV,\)où\(E = \big\{(x,y,z) \,|\, 0 \leq x \leq 2, \, 0 \leq y \leq 2x, \, 0 \leq z \leq 4 - x - y \big\}\)

    17. \(\displaystyle \iiint_E y \, dV,\)où\(E = \big\{(x,y,z) \,|\, -1 \leq x \leq 1, \, -\sqrt{1 - x^2} \leq y \leq \sqrt{1 - x^2}, \, 0 \leq z \leq 1 - x^2 - y^2 \big\}\)

    Réponse
    \(0\)

    18. \(\displaystyle \iiint_E x \, dV,\)où\(E = \big\{(x,y,z) \,|\, -2 \leq x \leq 2, \, -4\sqrt{1 - x^2} \leq y \leq \sqrt{4 - x^2}, \, 0 \leq z \leq 4 - x^2 - y^2 \big\}\)

    Dans les exercices 19 à 22, évaluez les intégrales triples sur la région délimitée\(E\) de la forme

    \(E = \big\{(x,y,z) \,|\, g_1 (y) \leq x \leq g_2(y), \, c \leq y \leq d, \, e \leq z \leq f \big\}\).

    19. \(\displaystyle \iiint_E x^2 \, dV,\)où\(E = \big\{(x,y,z) \,|\, 1 - y^2 \leq x \leq y^2 - 1, \, -1 \leq y \leq 1, \, 1 \leq z \leq 2 \big\}\)

    Réponse
    \(-\frac{64}{105}\)

    20. \(\displaystyle \iiint_E (y + \sin x) \, dV,\)où\(E = \big\{(x,y,z) \,|\, -y^4 \leq x \leq y^4, \, 0 \leq y \leq 2, \, 0 \leq z \leq 4\big\}\)

    21. \(\displaystyle \iiint_E (x - yz) \, dV,\)où\(E = \big\{(x,y,z) \,|\, -y^6 \leq x \leq \sqrt{y}, \, 0 \leq y \leq 1x, \, -1 \leq z \leq 1 \big\}\)

    Réponse
    \(\frac{11}{26}\)

    22. \(\displaystyle \iiint_E z \, dV,\)où\(E = \big\{(x,y,z) \,|\, 2 - 2y \leq x \leq 2 + \sqrt{y}, \, 0 \leq y \leq 1x, \, 2 \leq z \leq 3 \big\}\)

    Dans les exercices 23 à 26, évaluez les intégrales triples sur la région délimitée

    \(E = \big\{(x,y,z) \,|\, g_1(y) \leq x \leq g_2(y), \, c \leq y \leq d, \, u_1(x,y) \leq z \leq u_2 (x,y) \big\}\)

    23. \(\displaystyle \iiint_E z \, dV,\)où\(E = \big\{(x,y,z) \,|\, -y \leq x \leq y, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1 - x^4 - y^4 \big\}\)

    Réponse
    \(\frac{113}{450}\)

    24. \(\displaystyle \iiint_E (xz + 1) \, dV,\)où\(E = \big\{(x,y,z) \,|\, 0 \leq x \leq \sqrt{y}, \, 0 \leq y \leq 2, \, 0 \leq z \leq 1 - x^2 - y^2 \big\}\)

    25. \(\displaystyle \iiint_E (x - z) \, dV,\)où\(E = \big\{(x,y,z) \,|\, - \sqrt{1 - y^2} \leq x \leq y, \, 0 \leq y \leq \frac{1}{2}x, \, 0 \leq z \leq 1 - x^2 - y^2 \big\}\)

    Réponse
    \(\frac{1}{160}(6 \sqrt{3} - 41)\)

    26. \(\displaystyle \iiint_E (x + y) \, dV,\)où\(E = \big\{(x,y,z) \,|\, 0 \leq x \leq \sqrt{1 - y^2}, \, 0 \leq y \leq 1x, \, 0 \leq z \leq 1 - x \big\}\)

    Dans les exercices 27 à 30, évaluez les intégrales triples sur la région délimitée

    \(E = \big\{(x,y,z) \,|\, (x,y) \in D, \, u_1 (x,y) x \leq z \leq u_2 (x,y) \big\}\), où\(D\) est la\(E\) projection de sur le\(xy\) plan.

    27. \(\displaystyle \iint_D \left(\int_1^2 (x + y) \, dz \right) \, dA,\)où\(D = \big\{(x,y) \,|\, x^2 + y^2 \leq 1\big\}\)

    Réponse
    \(\frac{3\pi}{2}\)

    28. \(\displaystyle \iint_D \left(\int_1^3 x (z + 1)\, dz \right) \, dA,\)où\(D = \big\{(x,y) \,|\, x^2 -y^2 \geq 1, \, x \leq \sqrt{5}\big\}\)

    29. \(\displaystyle \iint_D \left(\int_0^{10-x-y} (x + 2z) \, dz \right) \, dA,\)où\(D = \big\{(x,y) \,|\, y \geq 0, \, x \geq 0, \, x + y \leq 10\big\}\)

    Réponse
    \(1250\)

    30. \(\displaystyle \iint_D \left(\int_0^{4x^2+4y^2} y \, dz \right) \, dA,\)où\(D = \big\{(x,y) \,|\, x^2 + y^2 \leq 4, \, y \geq 1, \, x \geq 0\big\}\)

    31. Le solide\(E\) délimité par\(y^2 + z^2 = 9, \, z = 0\) et\(x = 5\) est illustré dans la figure suivante. Évaluez l'intégrale\(\displaystyle \iiint_E z \, dV\) en intégrant d'abord par rapport à\(z\), puis\(y\), et ensuite\(x\).

    Forme arquée solide qui atteint son maximum le long de l'axe y avec z = 3. La forme atteint zéro à y = plus ou moins 3, et le graphique est tronqué à x = 0 et 5.

    Réponse
    \(\displaystyle \int_0^5 \int_{-3}^3 \int_0^{\sqrt{9-y^2}} z \, dz \, dy \, dx = 90\)

    32. The solid \(E\) bounded by \(y = \sqrt{x}, \, x = 4, \, y = 0\), and \(z = 1\) is given in the following figure. Evaluate the integral \(\displaystyle \iiint_E xyz \, dV\) by integrating first with respect to \(x\), then \(y\), and then \(z\).

    A quarter section of an oval cylinder with z from negative 2 to positive 1. The solid is bounded by y = 0 and x = 4, and the top of the shape runs from (0, 0, 1) to (4, 2, 1) in a gentle arc.

    33. [T] Le volume d'un solide\(E\) est donné par l'intégrale\(\displaystyle \int_{-2}^0 \int_x^0 \int_0^{x^2+y^2} \, dz \, dy \, dx.\) Utiliser un système d'algèbre informatique (CAS) pour représenter graphiquement\(E\) et trouver son volume. Arrondissez votre réponse à deux décimales.

    Réponse
    \(V \approx 5.33 \text{ units}^3\)

    Forme complexe qui commence à l'origine et atteint son maximum à (moins 2, moins 2, 8). La forme est tronquée par le plan x = y, le plan x = 0, le plan y = moins 2, le plan z = 0 et une forme triangulaire complexe avec des bords et des côtés incurvés (négatif 2, négatif 2, 8), (0, 0, 0, 0) et (0, négatif 2, 4).

    34. [T] Le volume d'un solide\(E\) is given by the integral \(\displaystyle \int_{-1}^0 \int_{-x^3}^0 \int_0^{1+\sqrt{x^2+y^2}} \, dz \, dy \, dx.\) Use a CAS to graph \(E\) and find its volume \(V\). Round your answer to two decimal places.

    In exercises 35 - 38, use two circular permutations of the variables \(x, \, y,\) and \(z\) to write new integrals whose values equal the value of the original integral. A circular permutation of \(x, \, y\), and \(z\) is the arrangement of the numbers in one of the following orders: \(y, \, z,\) and \(x\) or \(z, \, x,\) and \(y\).

    35. \(\displaystyle \int_0^1 \int_1^3 \int_2^4 (x^2z^2 + 1) \, dx \, dy \, dz\)

    Answer
    \(\displaystyle \int_0^1 \int_1^3 \int_2^4 (y^2z^2 + 1) \, dz \, dx \, dy;\)
    \(\displaystyle \int_0^1 \int_1^3 \int_2^4 (x^2y^2 + 1) \, dy \, dz \, dx\)

    36. \(\displaystyle \int_0^3 \int_0^1 \int_0^{-x+1} (2x + 5y + 7z) dy \, dx \, dz\)

    37. \(\displaystyle \int_0^1 \int_{-y}^y \int_0^{1-x^4-y^4} \ln x \, dz \, dx \, dy\)

    38. \(\displaystyle \int_{-1}^1 \int_0^1 \int_{-y^6}^{\sqrt{y}} (x + yz) \, dx \, dy \, dz\)

    39. Set up the integral that gives the volume of the solid \(E\) bounded by \(y^2 = x^2 + z^2\) and \(y = a^2\), where \(a > 0\).

    Answer
    \(\displaystyle V = \int_{-a}^a \int_{-\sqrt{a^2-z^2}}^{\sqrt{a^2-z^2}} \int_{\sqrt{x^2+z^2}}^{a^2} \, dy \, dx \, dz\)

    40. Set up the integral that gives the volume of the solid \(E\) bounded by \(x = y^2 + z^2\) and \(x = a^2\), where \(a > 0\).

    Average Value of a Function

    41. Find the average value of the function \(f(x,y,z) = x + y + z\) over the parallelepiped determined by \(x + 0, \, x = 1, \, y = 0, \, y = 3, \, z = 0\), and \(z = 5\).

    Answer
    \(\frac{9}{2}\)

    42. Find the average value of the function \(f(x,y,z) = xyz\) over the solid \(E = [0,1] \times [0,1] \times [0,1]\) situated in the first octant.

    Finding Volumes using Triple Integrals

    43. Find the volume of the solid \(E\) that lies under the plane \(x + y + z = 9\) and whose projection onto the \(xy\)-plane is bounded by \(x = sqrt{y-1}, \, x = 0\), and \(x + y = 7\).

    Answer
    \(\frac{156}{5} \text{ units}^3\)

    44. Find the volume of the solid \(E\) that lies under the plane \(2x+y+z=8\) and whose projection onto the \(xy\)-plane is bounded by \(x = sin^{-1} y, \, y = 0\), and \(x = \frac{\pi}{2}\).

    a. Show that the equations of the planes of the lateral faces of the pyramid are \(4y + z = 8, \, 4y - z = -8, \, 4x + z = 8\), and \(-4x + z = 8\).

    b. Find the volume of the pyramid.

    Answer
    a. Answers may vary;
    b. \(\frac{128}{3} \text{ units}^3\)

    46. Consider the pyramid with the base in the \(xy\)-plane of \([-3,3] \times [-3,3]\) and the vertex at the point \((0,0,9)\).

    a. Show that the equations of the planes of the side faces of the pyramid are \(3y + z = 9, \, 3y + z = 9, \, y = 0\) and \(x = 0\).

    b. Find the volume of the pyramid.

    47. The solid \(E\) bounded by the sphere of equation \(x^2 + y^2 + z^2 = r^2\) with \(r > 0\) and located in the first octant is represented in the following figure.

    The eighth of a sphere of radius 2 with center at the origin for positive x, y, and z.

    a. Écrivez la triple intégrale qui donne le volume de\(E\) en intégrant d'abord par rapport à\(z\), puis avec\(y\), puis avec\(x\).

    b. Réécrivez l'intégrale de la partie a. comme intégrale équivalente dans cinq autres ordres.

    Réponse

    a.\(\displaystyle \int_0^4 \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} \, dz \, dy \, dx\)
    b.\(\displaystyle \int_0^2 \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} \, dz \, dx \, dy,\)
    \(\displaystyle \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} \, dy \, dx \, dz,\)
    \(\displaystyle \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dy \, dz \, dx,\)
    \(\displaystyle \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dx \, dy \, dz,\)
    \(\displaystyle \int_0^r \int_0^{\sqrt{r^2-x^2}} \int_0^{\sqrt{r^2-x^2-y^2}} dx \, dz \,dy\)

    48. Le solide\(E\) délimité par la sphère d'équation\(9x^2 + 4y^2 + z^2 = 1\) et situé dans le premier octant est représenté sur la figure suivante.

    Dans le premier octant, on voit une forme complexe qui est à peu près un solide ovoïde dont le centre est l'origine, la hauteur 1, la largeur 0,5 et la longueur 0,35.

    a. Écrivez la triple intégrale qui donne le volume de\(E\) by integrating first with respect to \(z\) then with \(y\) and then with \(x\).

    b. Rewrite the integral in part a. as an equivalent integral in five other orders.

    49. Find the volume of the prism with vertices \((0,0,0), \, (2,0,0), \, (2,3,0), \, (0,3,0), \, (0,0,1)\), and \((2,0,1)\).

    Answer
    \(3 \text{ units}^3\)

    50. Find the volume of the prism with vertices \((0,0,0), \, (4,0,0), \, (4,6,0), \, (0,6,0), \, (0,0,1)\), and \((4,0,1)\).

    51. The solid \(E\) bounded by \(z = 10 - 2x - y\) and situated in the first octant is given in the following figure. Find the volume of the solid.

    A tetrahedron bounded by the x y, y z, and x z planes and a triangle with vertices (0, 0, 10), (5, 0, 0), and (0, 10, 0).

    Réponse
    \(\frac{250}{3} \text{ units}^3\)

    52. Le solide\(E\) délimité par\(z = 1 - x^2\) et situé dans le premier octant est représenté sur la figure suivante. Détermine le volume du solide.

    Forme complexe dans le premier octant avec une hauteur de 1, une largeur de 5 et une longueur de 1. La forme semble être un quart de cylindre légèrement déformé de rayon 1 et de largeur 5.

    Intégrales triples approximatives

    53. La règle du point médian pour la triple intégrale\(\displaystyle \iiint_B f(x,y,z) \,dV\) over the rectangular solid box \(B\) is a generalization of the midpoint rule for double integrals. The region \(B\) is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum \[\sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(\bar{x_i}, \bar{y_j}, \bar{z_k}) \Delta V,\nonumber \] where \((\bar{x_i}, \bar{y_j}, \bar{z_k})\) is the center of the box \(B_{ijk}\) and \(\Delta V\) is the volume of each subbox. Apply the midpoint rule to approximate \[\iiint_B x^2 \,dV\nonumber \] over the solid \(B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1 \big\}\) by using a partition of eight cubes of equal size. Round your answer to three decimal places.

    Answer
    \(\displaystyle \iiint_B f(x,y,z) \,dV\quad\) \(\approx\quad\frac{5}{16} \approx 0.313 \text{ units}^3\)

    54. [T]

    a. Apply the midpoint rule to approximate \(\displaystyle \iiint_B e^{-x^2} \, dV\) over the solid \(B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1 \big\}\) by using a partition of eight cubes of equal size. Round your answer to three decimal places.

    b. Use a CAS to improve the above integral approximation in the case of a partition of \(n^3\) cubes of equal size, where \(n = 3,\,4,\, ..., \,10\).

    Applications

    55. Suppose that the temperature in degrees Celsius at a point \((x,y,z)\) of a solid \(E\) bounded by the coordinate planes and the plane \(x + y + z = 5\) is given by: \[T (x,y,z) = xz + 5z + 10\nonumber \] Find the average temperature over the solid.

    Answer
    \(17.5^{\circ}\) C

    56. Suppose that the temperature in degrees Fahrenheit at a point \((x,y,z)\) of a solid \(E\) bounded by the coordinate planes and the plane \(x + y + z = 5\) is given by: \[T(x,y,z) = x + y + xy\nonumber \] Find the average temperature over the solid.

    57. Show that the volume of a right square pyramid of height \(h\) and side length \(a\) is \( v = \dfrac{ha^2}{3}\) by using triple integrals.

    58. Show that the volume of a regular right hexagonal prism of edge length \(a\) is \(\dfrac{3a^3 \sqrt{3}}{2}\) by using triple integrals.

    59. Show that the volume of a regular right hexagonal pyramid of edge length \(a\) is \(\dfrac{a^3 \sqrt{3}}{2}\) by using triple integrals.

    60. If the charge density at an arbitrary point \((x,y,z)\) of a solid \(E\) is given by the function \(\rho (x,y,z)\), then the total charge inside the solid is defined as the triple integral \(\displaystyle \iiint_E \rho (x,y,z) \,dV.\) Assume that the charge density of the solid \(E\) enclosed by the paraboloids \(x = 5 - y^2 - z^2\) and \(x = y^2 + z^2 - 5\) is equal to the distance from an arbitrary point of \(E\) to the origin. Set up the integral that gives the total charge inside the solid \(E\).

    Answer
    Total Charge inside the Solid \(E \quad=\quad\) \(\displaystyle \int_{-\sqrt{5}}^{\sqrt{5}}\int_{-\sqrt{5-y^2}}^{\sqrt{5-y^2}}\int_{y^2+z^2-5}^{5 - y^2 - z^2} \sqrt{x^2+y^2+z^2}\,dx\,dz\,dy\)