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15.8: Sura ya 15 Mazoezi ya Mapitio

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    Kweli au Uongo? Thibitisha jibu lako kwa ushahidi au mfano wa kukabiliana.

    1. \(\displaystyle ∫_a^b∫_c^d f(x,y) \, dy \, dx = ∫_c^d∫_a^b f(x,y) \, dy \, dx\)

    2. Theorem ya Fubini inaweza kupanuliwa kwa vipimo vitatu, kwa muda mrefu kama\(f\) inavyoendelea katika vigezo vyote.

    Jibu
    Kweli

    3. Muhimu\(\displaystyle ∫_0^{2π}∫_0^1∫_r^1 \,dz \, dr \, dθ\) unawakilisha kiasi cha koni sahihi.

    4. Jacobian ya mabadiliko kwa ajili\(x=u^2−2v, \, y=3v−2uv\) ya hutolewa na\(−4u^2+6u+4v.\)

    Jibu
    Uongo

    Tathmini integrals zifuatazo.

    5. \(\displaystyle \iint_R (5x^3y^2−y^2) \, dA,\)wapi\(R=\big\{(x,y) \,|\, 0≤x≤2,\, 1≤y≤4\big\}\)

    6. \(\displaystyle \iint_D \dfrac{y}{3x^2+1} \, dA,\)wapi\( D=\big\{(x,y) \,|\, 0≤x≤1, \, −x≤y≤x\big\}\)

    Jibu
    \(0\)

    7. \(\displaystyle \iint_D \sin(x^2+y^2) \, dA\)\(D\)wapi disk ya radius\(2\) iliyozingatia asili.

    8. \(\displaystyle \int_0^1\int_y^1 xye^{x^2}\,dx \, dy\)

    Jibu
    \(\frac{1}{4}\)

    9. \(\displaystyle \int_{−1}^1\int_0^z\int_0^{x−z} 6 \, dy \, dx\, dz\)

    10. \(\displaystyle \iiint_R 3y \, dV,\)wapi\(R=\big\{(x,y,z) \,|\, 0≤x≤1, \, 0≤y≤x, \, 0≤z≤9−y^2\big\}\)

    Jibu
    \(1.475\)

    11. \(\displaystyle \int_0^2\int_0^{2π}\int_r^1 r \, dz \, dθ \, dr\)

    12. \(\displaystyle \int_0^{2π}\int_0^{π/2}\int_1^3 ρ^2\sin(φ) \, dρ \, dφ \, dθ\)

    Jibu
    \(\frac{52\pi}{3}\)

    13. \(\displaystyle \int_0^1\int_{−\sqrt{1−x^2}}^{\sqrt{1−x^2}}\int_{−\sqrt{1−x^2−y^2}}^{\sqrt{1−x^2−y^2}} \, dz \, dy \, dx\)

    Kwa matatizo yafuatayo, tafuta eneo maalum au kiasi.

    14. Eneo la mkoa iliyoambatanishwa na petal moja ya\(r=\cos(4θ).\)

    Jibu
    \(\frac{\pi}{16} \text{ units}^3\)

    15. Kiasi cha imara kilicho kati ya paraboloid\(z=2x^2+2y^2\) na ndege\(z=8.\)

    16. Kiasi cha imara imefungwa na silinda\(x^2+y^2=16\) na kutoka\(z=1\)\(z+x=2.\)

    Jibu
    \(93.291 \text{ units}^3\)

    17. Kiasi cha makutano kati ya nyanja mbili\(1,\) za radius, juu ambayo katikati yake ni\((0,\,0,\,0.25)\) chini, ambayo inazingatia\((0,\,0,\,0).\)

    Kwa matatizo yafuatayo, pata katikati ya wingi wa kanda.

    18. \(ρ(x,y)=xy\)kwenye mduara na radius\(1\) katika roboduara ya kwanza tu.

    Jibu
    \( \left( \frac{8}{15}, \, \frac{8}{15} \right) \)

    19. \(ρ(x,y)=(y+1)\sqrt{x}\)katika mkoa umepakana\(y=e^x, \, y=0,\) na\(x=1.\)

    20. \(ρ(x,y,z)=z\)kwenye koni iliyoingizwa na radius\(2\) na urefu\(2.\)

    Jibu
    \( \left( 0, \, 0, \, \frac{8}{5} \right) \)

    21. Kiasi - koni ya ice cream ambayo hutolewa na imara hapo juu\(z=\sqrt{x^2+y^2}\) na chini.\(z^2+x^2+y^2=z.\)

    matatizo yafuatayo kuchunguza Mount Holly katika jimbo la Michigan. Mlima Holly ni taka kwamba alikuwa waongofu katika mapumziko Ski. Sura ya Mlima Holly inaweza kuhesabiwa na koni ya mviringo ya kulia ya urefu wa 1100 ft na radius 6000 ft.

    22. Kama takataka iliyounganishwa kutumika kujenga Mlima Holly kwa wastani ina wiani\(400\text{ lb/ft}^3,\) kupata kiasi cha kazi zinazohitajika kujenga mlima.

    Jibu
    \(1.452\pi \times 10^{15}\)ft-lb

    23. Kwa kweli, kuna uwezekano mkubwa kwamba takataka chini ya Mlima Holly imekuwa zaidi kuunganishwa na uzito wote wa takataka hapo juu. Fikiria kazi ya wiani kwa heshima na urefu: wiani juu ya mlima bado ni wiani\(400\text{ lb/ft}^3,\) na wiani huongezeka. Kila miguu 100 zaidi, wiani mara mbili. Uzito wa jumla wa Mlima Holly ni nini?

    Matatizo yafuatayo yanazingatia halijoto na wiani wa tabaka za Dunia.

    24. [T] Joto la tabaka za Dunia linaonyeshwa katika jedwali hapa chini. Tumia calculator yako ili kufaa polynomial ya shahada 3 hadi joto pamoja na radius ya Dunia. Kisha pata joto la wastani la Dunia. (Kidokezo: kuanza saa 0 katika msingi wa ndani na ongezeko nje kuelekea uso)

    Chanzo: http://www.enchantedlearning.com/sub...h/Inside.shtml
    Tabaka Urefu kutoka katikati (km) Joto °C
    rocky ukanda 0 hadi 40 0
    Mantle ya juu 40 hadi 150 870
    Mantle 400 hadi 650 870
    Mantel ya ndani 650 hadi 2700 870
    kuyeyuka nje Core 2890 kwa 5150 4300
    Msingi wa Ndani 5150 kwa 6378 7200
    Jibu
    \(y=−1.238×10^{−7}x^3+0.001196x^2−3.666x+7208\);
    Halijoto wastani ni takriban 2800 °C.

    25. [T] Uzito wa tabaka za Dunia huonyeshwa katika jedwali hapa chini. Kutumia calculator yako au programu ya kompyuta, kupata bora fit quadratic equation kwa wiani. Kutumia equation hii, pata molekuli jumla ya Dunia.

    Chanzo: http://hyperphysics.phy-astr.gsu.edu...rthstruct.html
    Tabaka Urefu kutoka katikati (km) Uzito wiani (\(\text{g/cm}^3\))
    Msingi wa Ndani 0 \ (\ maandishi {g/cm} ^3\))” data-valign="top">12.95
    Msingi wa nje 1228 \ (\ maandishi {g/cm} ^3\))” data-valign="top">11.05
    Mantle 3488 \ (\ maandishi {g/cm} ^3\))” data-valign="top">5.00
    Mantle ya juu 6338 \ (\ maandishi {g/cm} ^3\))” data-valign="top">3.90
    Ukoko 6378 \ (\ maandishi {g/cm} ^3\))” data-valign="top">2.55

    Matatizo yafuatayo yanahusu Theorem ya Pappus (angalia Moments na Vituo vya Misa kwa ajili ya kufufua), njia ya kuhesabu kiasi kwa kutumia centroids. Kutokana kanda\(R,\) wakati unazunguka\(x\) -mhimili kiasi kinachotolewa\(V_x=2πA\overline{y},\) na na wakati unazunguka\(y\) -mhimili kiasi kinatolewa na\(V_y=2πA\overline{x},\) wapi\(A\) eneo la\(R.\) Fikiria eneo lililofungwa na\(x^2+y^2=1\) na hapo juu\(y=x+1.\)

    26. Find kiasi wakati zinahusu kanda karibu\(x\) mhimili -.

    Jibu
    \(\frac{\pi}{3} \text{ units}^3\)

    27. Find kiasi wakati zinahusu kanda karibu\(y\) mhimili -.