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16.6E: Mazoezi ya Sehemu ya 16.6

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    178912

    • Edwin “Jed” Herman & Gilbert Strang
    • OpenStax
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    Katika mazoezi ya 1 - 4, onyesha kama taarifa ni za kweli au za uongo.

    1. Ikiwa uso\(S\) hutolewa na\(\{(x,y,z) : \, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, z = 10 \}\), basi\(\displaystyle \iint_S f(x,y,z) \, dS = \int_0^1 \int_0^1 f (x,y,10) \, dx \, dy.\)

    Jibu
    Kweli

    2. Ikiwa uso\(S\) hutolewa na\(\{(x,y,z) : \, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, z = x \}\), basi\(\displaystyle \iint_S f(x,y,z) \, dS = \int_0^1 \int_0^1 f (x,y,x) \, dx \, dy.\)

    3. Surface\(\vecs r = \langle v \, \cos u, \, v \, \sin u, \, v^2 \rangle,\) kwa\( 0 \leq u \leq \pi, \, 0 \leq v \leq 2\) ni uso sawa\(\vecs r = \langle \sqrt{v} \, \cos 2u, \, \sqrt{v} \, \sin 2u, \, v \rangle,\) kwa\( 0 \leq u \leq \dfrac{\pi}{2}, \, 0 \leq v \leq 4\).

    Jibu
    Kweli

    4. Kutokana na parameterization ya kawaida ya nyanja, vectors kawaida\(t_u \times t_v\) ni vectors nje ya kawaida.

    Katika mazoezi ya 5 - 10, pata maelezo ya parametric kwa nyuso zifuatazo.

    5. Ndege\(3x - 2y + z = 2\)

    Jibu
    \(\vecs r(u,v) = \langle u, \, v, \, 2 - 3u + 2v \rangle \)kwa\(-\infty \leq u < \infty\) na\( - \infty \leq v < \infty\).

    6. Paraboloid\(z = x^2 + y^2\), kwa\(0 \leq z \leq 9\).

    7. Ndege\(2x - 4y + 3z = 16\)

    Jibu
    \(\vecs r(u,v) = \langle u, \, v, \, \dfrac{1}{3} (16 - 2u + 4v) \rangle \)kwa\(|u| < \infty\) na\(|v| < \infty\).

    8. Frustum ya mbegu\(z^2 = x^2 + y^2\), kwa\(2 \leq z \leq 8\)

    9. Sehemu ya silinda\(x^2 + y^2 = 9\) katika octant kwanza, kwa\(0 \leq z \leq 3\)

    Mchoro katika vipimo vitatu vya sehemu ya silinda na radius 3. Katikati ya juu yake ya mviringo ni (0,0,3). Sehemu ipo kwa x, y, na z kati ya 0 na 3.

    Jibu
    \(\vecs r(u,v) = \langle 3 \, \cos u, \, 3 \, \sin u, \, v \rangle \) for \(0 \leq u \leq \dfrac{\pi}{2}, \, 0 \leq v \leq 3\)

    10. A cone with base radius \(r\) and height \(h,\) where \(r\) and \(h\) are positive constants.

    For exercises 11 - 12, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface.

    11. [T] Half cylinder \(\{ (r, \theta, z) : \, r = 4, \, 0 \leq \theta \leq \pi, \, 0 \leq z \leq 7 \}\)

    Answer
    \(A = 87.9646\)

    12. [T] Plane \(z = 10 - z - y\) above square \(|x| \leq 2, \, |y| \leq 2\)

    In exercises 13 - 15, let \(S\) be the hemisphere \(x^2 + y^2 + z^2 = 4\), with \(z \geq 0\), and evaluate each surface integral, in the counterclockwise direction.

    13. \(\displaystyle \iint_S z\, dS\)

    Answer
    \(\displaystyle \iint_S z \, dS = 8\pi\)

    14. \(\displaystyle \iint_S (x - 2y) \, dS\)

    15. \(\displaystyle \iint_S (x^2 + y^2) \, dS\)

    Answer
    \(\displaystyle \iint_S (x^2 + y^2) \, dS = 16 \pi\)

    In exercises 16 - 18, evaluate \(\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS\) for vector field \(\vecs F\) where \(\vecs N\) is an outward normal vector to surface \(S.\)

    16. \(\vecs F(x,y,z) = x\,\mathbf{\hat i}+ 2y\,\mathbf{\hat j} = 3z\,\mathbf{\hat k}\), and \(S\) is that part of plane \(15x - 12y + 3z = 6\) that lies above unit square \(0 \leq x \leq 1, \, 0 \leq y \leq 1\).

    17. \(\vecs F(x,y) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j}\), and \(S\) is hemisphere \(z = \sqrt{1 - x^2 - y^2}\).

    Answer
    \(\displaystyle \iint_S \vecs F \cdot \vecs N \, dS = \dfrac{4\pi}{3}\)

    18. \(\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2\,\mathbf{\hat j} + z^2\,\mathbf{\hat k}\), and \(S\) is the portion of plane \(z = y + 1\) that lies inside cylinder \(x^2 + y^2 = 1\).

    A cylinder and an intersecting plane shown in three-dimensions. S is the portion of the plane z = y + 1 inside the cylinder x^2 + y ^2 = 1.

    Katika mazoezi 19 - 20, takriban wingi wa lamina yenye homogeneous ambayo ina sura ya uso uliopewa\(S.\) Pande zote hadi sehemu nne za decimal.

    19. [T]\(S\) ni uso\(z = 4 - x - 2y\), na\(z \geq 0, \, x \geq 0, \, y \geq 0; \, \xi = x.\)

    Jibu
    \(m \approx 13.0639\)

    20. [T]\(S\) ni uso\(z = x^2 + y^2\), kwa\(z \leq 1; \, \xi = z\).

    21. [T]\(S\) ni uso\(x^2 + y^2 + x^2 = 5\), kwa\(z \geq 1; \, \xi = \theta^2\).

    Jibu
    \(m \approx 228.5313\)

    22. Tathmini\(\displaystyle \iint_S (y^2 z\,\mathbf{\hat i}+ y^3\,\mathbf{\hat j} + xz\,\mathbf{\hat k}) \cdot dS,\)\(S\) wapi uso wa mchemraba\(-1 \leq x \leq 1, \, -1 \leq y \leq 1\), na\(0 \leq z \leq 2\) kwa mwelekeo wa kinyume.

    23. Kutathmini uso muhimu\(\displaystyle \iint_S g \, dS,\) ambapo\(g(x,y,z) = xz + 2x^2 - 3xy\) na\(S\) ni sehemu ya ndege\(2x - 3y + z = 6\) ambayo ipo juu ya kitengo mraba\(R: 0 \leq x \leq 1, \, 0 \leq y \leq 1\).

    Jibu
    \(\displaystyle \iint_S g\,dS = 3 \sqrt{4}\)

    24. Tathmini\(\displaystyle \iint_S (x + y + z)\, dS,\) wapi uso\(S\) hufafanuliwa parametrically na\(\vecs R(u,v) = (2u + v)\,\mathbf{\hat i} + (u - 2v)\,\mathbf{\hat j} + (u + 3v)\,\mathbf{\hat k}\) kwa\(0 \leq u \leq 1\), na\(0 \leq v \leq 2\).

    Mchoro wa tatu-dimensional wa uso uliopewa, unaoonekana kuwa ndege yenye kutembea kwa njia ya ndege (x, y).

    25. [T] Tathmini\(\displaystyle \iint_S (x - y^2 + z)\, dS,\) where \(S\) is the surface defined parametrically by \(\vecs R(u,v) = u^2\,\mathbf{\hat i} + v\,\mathbf{\hat j} + u\,\mathbf{\hat k}\) for \(0 \leq u \leq 1, \, 0 \leq v \leq 1\).

    A three-dimensional diagram of the given surface, which appears to be a curve with edges parallel to the y-axis. It increases in x components and decreases in z components the further it is from the y axis.

    Jibu
    \(\displaystyle \iint_S (x^2 + y - z) \, dS \approx 0.9617\)

    26. [T] Kutathmini wapi uso\(S\) inavyoelezwa na\(\vecs R(u,v) = u\,\mathbf{\hat i} - u^2\,\mathbf{\hat j} + v\,\mathbf{\hat k}, \, 0 \leq u \leq 2, \, 0 \leq v \leq 1\) kwa\(0 \leq u \leq 1, \, 0 \leq v \leq 2\).

    27. Tathmini\(\displaystyle \iint_S (x^2 + y^2) \, dS,\)\(S\) wapi uso umefungwa juu ya hemisphere\(z = \sqrt{1 - x^2 - y^2}\), na chini kwa ndege\(z = 0\).

    Jibu
    \(\displaystyle \iint_S (x^2 + y^2) \, dS = \dfrac{4\pi}{3}\)

    28. Tathmini\(\displaystyle \iint_S (x^2 + y^2 + z^2) \, dS,\) ambapo\(S\) ni sehemu ya ndege ambayo ipo ndani ya silinda\(x^2 + y^2 = 1\).

    29. [T] Kutathmini\(\displaystyle \iint_S x^2 z \, dS,\) ambapo\(S\) ni sehemu ya koni\(z^2 = x^2 + y^2\) uongo kati ya ndege\(z = 1\) na\(z = 4\).

    Mchoro wa koni iliyotolewa juu ya ufunguzi katika vipimo vitatu. Koni hukatwa na ndege z=1 na z=4.

    Jibu
    \(\text{div}\,\vecs F = a + b\)

    \(\displaystyle \iint_S x^2 zdS = \dfrac{1023\sqrt{2\pi}}{5}\)

    30. [T] Evaluate \(\displaystyle \iint_S \frac{xz}{y} \, dS,\) where \(S\) is the portion of cylinder \(x = y^2\) that lies in the first octant between planes \(z = 0, \, z = 5\), and \(y = 4\).

    A diagram of the given cylinder in three-dimensions. It is cut by the planes z=0, z=5, y=1, and y=4.

    31. [T] Tathmini\(\displaystyle \iint_S (z + y) \, dS,\) ambapo\(S\) ni sehemu ya graph ya\( z = \sqrt{1 - x^2}\) katika octant kwanza kati ya\(xy\) -ndege na ndege\(y = 3\).

    Mchoro wa uso uliotolewa katika vipimo vitatu katika octant ya kwanza kati ya ndege ya xz na ndege y=3. Grafu iliyotolewa ya z = mizizi ya mraba ya (1-x ^ 2) inaweka chini ya concave chini ya Curve kutoka pamoja (0, y,1) hadi pamoja (1, y,0). Inaonekana kama sehemu ya silinda ya usawa na msingi pamoja na ndege ya xz na urefu kando ya mhimili y.

    Jibu
    \(\displaystyle \iint_S (z + y) \, dS \approx 10.1\)

    32. Evaluate \(\displaystyle \iint_S xyz\, dS\) if \(S\) is the part of plane \(z = x + y\) that lies over the triangular region in the \(xy\)-plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).

    33. Find the mass of a lamina of density \(\xi (x,y,z) = z\) in the shape of hemisphere \(z = (a^2 - x^2 - y^2)^{1/2}\).

    Answer
    \(m = \pi a^3\)

    34. Compute \(\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS,\) where \(\vecs F(x,y,z) = x\,\mathbf{\hat i} - 5y\,\mathbf{\hat j} + 4z\,\mathbf{\hat k}\) and \(\vecs N\) is an outward normal vector \(S,\) where \(S\) is the union of two squares \(S_1\) : \(x = 0, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1\) and \(S_2 \, : \, x = 0, \, 0 \leq x \leq 1, \, 0 \leq y \leq 1\).

    A diagram in three dimensions. It shows the square formed by the components x=0, 0 <= y <= 1, and 0 <= z <= 1. It also shows the square formed by the components z=1, 0 <= x <= 1, and 0 <= y <= 1.

    35. Compute\(\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS,\) wapi\(\vecs F(x,y,z) = xy\,\mathbf{\hat i} + z\,\mathbf{\hat j} + (x + y)\,\mathbf{\hat k}\) na\(\vecs N\) ni vector nje ya kawaida\(S,\) ambapo\(S\) ni mkoa triangular kukatwa kutoka ndege\(x + y + z = 1\) na chanya kuratibu shoka.

    Jibu
    \(\displaystyle \iint_S \vecs F \cdot \vecs N \, dS = \dfrac{13}{24}\)

    36. Compute\(\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS,\) wapi\(\vecs F(x,y,z) = 2yz\,\mathbf{\hat i} + (\tan^{-1}xz)\,\mathbf{\hat j} + e^{xy}\,\mathbf{\hat k}\) na\(\vecs N\) ni vector nje ya kawaida\(S,\) ambapo\(S\) ni uso wa nyanja\(x^2 + y^2 + z^2 = 1\).

    37. Compute\(\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS,\) wapi\(\vecs F(x,y,z) = xyz\,\mathbf{\hat i} + xyz\,\mathbf{\hat j} + xyz\,\mathbf{\hat k}\) na\(\vecs N\) ni nje ya kawaida vector\(S,\) ambapo\(S\) ni uso wa nyuso tano ya kitengo mchemraba\(0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1\) kukosa\(z = 0\).

    Jibu
    \(\displaystyle \iint_S \vecs F \cdot \vecs N \, dS = \dfrac{3}{4}\)

    Kwa ajili ya mazoezi 38 - 39, kueleza uso muhimu kama iterated mara mbili muhimu kwa kutumia makadirio\(S\) juu ya\(yz\) -ndege.

    38. \(\displaystyle \iint_S xy^2 z^3 \, dS;\)\(S\)ni sehemu ya kwanza ya octant ya ndege\(2x + 3y + 4z = 12\).

    39. \(\displaystyle \iint_S (x^2 - 2y + z) \, dS;\)\(S\)ni sehemu ya grafu ya\(4x + y = 8\) imepakana na ndege kuratibu na ndege\(z = 6\).

    Jibu
    \(\displaystyle \int_0^8 \int_0^6 \left( 4 - 3y + \dfrac{1}{16} y^2 + z \right) \left(\dfrac{1}{4} \sqrt{17} \right) \, dz \, dy\)

    Kwa ajili ya mazoezi 40 - 41, kueleza uso muhimu kama iterated mara mbili muhimu kwa kutumia makadirio\(S\) juu ya\(xz\) -ndege.

    40. \(\displaystyle \iint_S xy^2z^3 \, dS;\)\(S\)ni sehemu ya kwanza ya octant ya ndege\(2x + 3y + 4z = 12\).

    41. \(\displaystyle \iint_S (x^2 - 2y + z) \, dS;\)ni sehemu ya grafu ya\(4x + y = 8\) imepakana na ndege kuratibu na ndege\(z = 6\).

    Jibu
    \(\displaystyle \int_0^2 \int_0^6 \big[x^2 - 2 (8 - 4x) + z\big] \sqrt{17} \, dz \, dx\)

    42. Kutathmini uso muhimu\(\displaystyle \iint_S yz \, dS,\) ambapo\(S\) ni sehemu ya kwanza octant ya ndege\(x + y + z = \lambda\), ambapo\(\lambda\) ni mara kwa mara chanya.

    43. Tathmini ya uso muhimu\(\displaystyle \iint_S (x^2 z + y^2 z) \, dS,\) ambapo\(S\) ni hemisphere\(x^2 + y^2 + z^2 = a^2, \, z \geq 0.\)

    Jibu
    \(\displaystyle \iint_S (x^2 z + y^2 z) \, dS = \dfrac{\pi a^5}{2}\)

    44. Kutathmini uso muhimu\(\displaystyle \iint_S z \, dA,\) ambapo\(S\) ni uso\(z = \sqrt{x^2 + y^2}, \, 0 \leq z \leq 2\).

    45. Kutathmini uso muhimu\(\displaystyle \iint_S x^2 yz \, dS,\) ambapo\(S\) ni sehemu ya ndege\(z = 1 + 2x + 3y\) ambayo ipo juu ya mstatili\(0 \leq x \leq 3\) na\(0 \leq y \leq 2\).

    Jibu
    \(\displaystyle \iint_S x^2 yz \, dS = 171 \sqrt{14}\)

    46. Kutathmini uso muhimu\(\displaystyle \iint_S yz \, dS,\) ambapo\(S\) ni ndege\(x + y + z = 1\) ambayo ipo katika octant kwanza.

    47. Kutathmini uso muhimu\(\displaystyle \iint_S yz \, dS,\) ambapo\(S\) ni sehemu ya ndege\(z = y + 3\) ambayo ipo ndani ya silinda\(x^2 + y^2 = 1\).

    Jibu
    \(\displaystyle \iint_S yz \, dS = \dfrac{\sqrt{2}\pi}{4}\)

    Kwa mazoezi 48 - 50, tumia hoja za kijiometri ili kutathmini uingizaji wa uso uliotolewa.

    48. \(\displaystyle \iint_S \sqrt{x^2 + y^2 + z^2} \, dS,\)\(S\)wapi uso\(x^2 + y^2 + z^2 = 4, \, z \geq 0\)

    49. \(\displaystyle \iint_S (x\,\mathbf{\hat i} + y\,\mathbf{\hat j}) \cdot dS,\)ambapo\(S\) ni uso\(x^2 + y^2 = 4, \, 1 \leq z \leq 3\), oriented na wadudu kitengo kawaida akizungumzia nje

    Jibu
    \(\displaystyle \iint_S (x\,\mathbf{\hat i} + y\,\mathbf{\hat j}) \cdot dS = 16 \pi\)

    50. \(\displaystyle \iint_S (z\,\mathbf{\hat k}) \cdot dS,\)\(S\)wapi disc\(x^2 + y^2 \leq 9\) juu ya ndege\(z = 4\) oriented na kitengo wadudu kawaida akizungumzia zaidi

    51. Lamina ina sura ya sehemu ya nyanja\(x^2 + y^2 + z^2 = a^2\) iliyo ndani ya koni\(z = \sqrt{x^2 + y^2}\). Hebu\(S\) kuwa shell spherical unaozingatia katika asili na radius a, na\(C\) iwe sahihi mviringo koni na vertex katika asili na mhimili wa ulinganifu ambayo sanjari na\(z\) -axis. Kuamua wingi wa lamina ikiwa\(\delta(x,y,z) = x^2 y^2 z\).

    Mchoro katika vipimo vitatu. Koni inafungua juu na uhakika katika asili na asic ya ulinganifu ambayo inafanana na mhimili wa z. Nusu ya juu ya hemphere iliyo na kituo cha asili hufungua chini na imekatwa na ndege ya xy.

    Jibu
    \(m = \dfrac{\pi a^7}{192}\)

    52. A lamina has the shape of a portion of sphere \(x^2 + y^2 + z^2 = a^2\) that lies within cone \(z = \sqrt{x^2 + y^2}\). Let \(S\) be the spherical shell centered at the origin with radius a, and let \(C\) be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Suppose the vertex angle of the cone is \(\phi_0\), with \(0 \leq \phi_0 < \dfrac{\pi}{2}\). Determine the mass of that portion of the shape enclosed in the intersection of \(S\) and \(C.\) Assume \(\delta(x,y,z) = x^2y^2z.\)

    A diagram in three dimensions. A cone opens upward with point at the origin and an asic of symmetry that coincides with the z-axis. The upper half of a hemisphere with center at the origin opens downward and is cut off by the xy-plane.

    53. Kikombe cha karatasi kina sura ya koni ya mviringo ya mviringo ya urefu wa 6 katika. na radius ya juu 3 ndani. Ikiwa kikombe kinajaa uzito wa maji\(62.5 \, lb/ft^3\), pata ukubwa wa nguvu ya jumla inayotumiwa na maji kwenye uso wa ndani wa kikombe.

    Jibu
    \(F \approx 4.57 \, lb\)

    Kwa mazoezi 54 - 55, shamba la vector la mtiririko wa joto kwa ajili ya kufanya vitu i\(\vecs F = - k\vecs\nabla T\),\(T(x,y,z)\) wapi joto katika kitu na\(k > 0\) ni mara kwa mara ambayo inategemea nyenzo. Pata mtiririko wa nje wa\(\vecs F\) nyuso zifuatazo\(S\) kwa mgawanyo wa joto uliotolewa na kudhani\(k = 1\).

    54. \(T(x,y,z) = 100 e^{-x-y}\);\(S\) lina nyuso za mchemraba\(|x| \leq 1, \, |y| \leq 1, \, |z| \leq 1\).

    55. \(T(x,y,z) = - \ln (x^2 + y^2 + z^2)\);\(S\) ni nyanja\(x^2 + y^2 + z^2 = a^2\).

    Jibu
    \(8\pi a\)

    Kwa mazoezi 56 - 57, fikiria mashamba ya radial\(\vecs F = \dfrac{\langle x,y,z \rangle}{(x^2+y^2+z^2)^{\dfrac{p}{2}}} = \dfrac{r}{|r|^p}\),\(p\) wapi idadi halisi. Hebu\(S\) iwe\(A\) na\(B\) nyanja na kuzingatia asili na radii\(0 < a < b\). Jumla ya nje ya flux kote\(S\) ina flux nje katika nyanja ya nje,\(B\) chini ya flux ndani\(S\) ya nyanja ya ndani.\(A.\)

    Mchoro katika vipimo vitatu vya nyanja mbili, moja yaliyomo ndani ya nyingine. Vituo vyao vyote ni asili. Mishale inaelezea kuelekea asili kutoka nje ya nyanja zote mbili.

    56. Pata jumla ya flux kote

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