Mafunso ndi Zitsanzo za Ma Integral Applications
Kuphatikizana ndi lingaliro lofunikira mu calculus lomwe limagwiritsidwa ntchito m'magawo osiyanasiyana a sayansi, monga fizikisi, zachuma, zamoyo, ndi uinjiniya. Ma Integrals amagwiritsidwa ntchito kuwerengera dera lomwe lili pansi pa curve, voliyumu ya chinthu cholimba, ntchito, kupanikizika, ndi zina zambiri. M'nkhaniyi, tikambirana zitsanzo zingapo za ntchito zophatikizana, kutsatiridwa ndi kufotokozera mwatsatanetsatane momwe tingazithetsere.
1. Kudziwa Malo Omwe Ali Pansi pa Khoma
Chimodzi mwa ntchito zofala kwambiri za ma integrals ndikuwerengera dera lomwe lili pansi pa curve ya ntchito pa nthawi inayake. Tiyerekeze kuti tikufuna kupeza dera la dera lomwe lili ndi curve \(y = x^2\) ndi axis \(x\) kuyambira \(x = 0\) mpaka \(x = 2\).
Chitsanzo cha mavuto:
Dziwani malo omwe ali pansi pa curve \(y = x^2\) kuyambira \(x = 0\) mpaka \(x = 2\).
Kukambirana:
Kuti tipeze dera lomwe lili pansi pa curve \(y = x^2\) kuchokera \(x = 0\) mpaka \(x = 2\), tifunika kuwerengera integral yeniyeni ya ntchitoyo:
\[ \int_{0}^{2} x^2 \, dx \]
Gawo 1: Dziwani chinthu chofunikira cha \(x^2\).
Dziwani kuti mfundo yofunika kwambiri ya \(x^2\) ndi:
\[ \int x^2 \, dx = \frac{x^3}{3} + C \]
Gawo 2: Ikani malire ofunikira \(0\) ku \(2\).
\[ \int_{0}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{2} \]
Gawo 3: Werengani mtengo wokwanira.
\[ \left. \frac{x^3}{3} \right|_{0}^{2} = \frac{2^3}{3} – \frac{0^3}{3} = \frac{8}{3} – 0 = \frac{8}{3} \]
Kotero, dera lomwe lili pansi pa curve \(y = x^2\) kuyambira \(x = 0\) mpaka \(x = 2\) ndi \( \frac{8}{3} \) mayunitsi a dera.
2. Kuwerengera Volume ya Zinthu Zozungulira
Ma Integrals amagwiritsidwanso ntchito kuwerengera voliyumu ya zinthu zolimba zozungulira. Ngati dera likuzunguliridwa mozungulira mzere wa \(x\), voliyumu ya chinthucho ingapezeke pogwiritsa ntchito njira ya disk kapena njira ya ring.
Chitsanzo cha mavuto:
Werengerani voliyumu ya chinthu chomwe chapangidwa pamene dera lomwe lili ndi curve \(y = \sqrt{x}\) ndi mzere \(x = 4\) zikuzunguliridwa mozungulira mzere wa \(x\).
Kukambirana:
Kuti tipeze voliyumu ya chinthu cholimba chozungulira, tingagwiritse ntchito njira ya disk. Voliyumu \(V\) ya chinthu cholimba chomwe chatulukapo ikhoza kufotokozedwa motere:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
Kumene \(f(x) = \sqrt{x}\), \(a = 0\), ndi \(b = 4\).
Gawo 1: Pangani voliyumu yofunikira.
\[ V = \pi \int_{0}^{4} (\sqrt{x})^2 \, dx \]
Gawo 2: Fewetsani ntchito mu integral.
\[ V = \pi \int_{0}^{4} x \, dx \]
Gawo 3: Dziwani chinthu chofunikira cha \(x\).
\[ \int x \, dx = \frac{x^2}{2} + C \]
Gawo 4: Ikani malire \(0\) ku \(4\).
\[ V = \pi \kumanzere[ \frac{x^2}{2} \kumanja]_{0}^{4} \]
Gawo 5: Werengani mtengo wokwanira.
\[ \left. \frac{x^2}{2} \right|_{0}^{4} = \pi \left( \frac{4^2}{2} – \frac{0^2}{2} \right) = \pi \left( \frac{16}{2} \right) = 8\pi \]
Kotero, voliyumu ya chinthu chomwe chatuluka ndi mayunitsi a voliyumu a \(8\pi\).
3. Kuwerengera Ntchito Yochitidwa ndi Mphamvu Yosinthasintha
Ntchito zophatikizana zimapezekanso mu fizikisi, imodzi mwa izo ndi kuwerengera ntchito yochitidwa ndi mphamvu yosinthasintha pamene chinthu chikuyenda kuchokera pamalo ena kupita kwina.
Chitsanzo cha mavuto:
Mphamvu \(F(x) = 3x^2\) Newton imagwira ntchito pa tinthu tomwe tikuyenda kuchokera pa \(x = 1\) mita kupita pa \(x = 3\) mita. Werengani ntchito yomwe mphamvuyo yachita.
Kukambirana:
Ntchito \(W\) yochitidwa ndi mphamvu \(F(x)\) ingapezeke powerengera integral ya \(F(x)\) pa kusuntha kuchokera \(a\) kupita ku \(b\):
\[ W = \int_{a}^{b} F(x) \, dx \]
Kumene \(a = 1\), \(b = 3\), ndi \(F(x) = 3x^2\).
Gawo 1: Pangani gawo lofunika kwambiri la ntchitoyo.
\[ W = \int_{1}^{3} 3x^2 \, dx \]
Gawo 2: Dziwani chinthu chofunikira cha \(3x^2\).
\[ \int 3x^2 \, dx = 3 \left( \frac{x^3}{3} \right) = x^3 + C \]
Gawo 3: Ikani malire \(1\) ku \(3\).
\[ W = \kumanzere[ x^3 \kumanja]_{1}^{3} \]
Gawo 4: Werengani mtengo wokwanira.
\[ W = \kumanzere. x^3 \kumanja|_{1}^{3} = 3^3 – 1^3 = 27 – 1 = 26 \]
Kotero, ntchito yochitidwa ndi mphamvu ndi \(26\) joules.
4. Kudziwa Kupanikizika kwa Madzi
Mu fizikisi, ma integrals amagwiritsidwanso ntchito kuwerengera kuthamanga kwa hydrostatic pamwamba pomizidwa mu madzi.
Chitsanzo cha mavuto:
Mbale yoyima yotalika mamita 6 ndi m'lifupi mamita 4 imamizidwa m'madzi ndipo pamwamba pake pamakhala pamwamba pa madzi. Werengani mphamvu yonse ya kuthamanga kwa madzi pa mbaleyo.
Kukambirana:
Kupanikizika pa kuya \(h\) m'madzi kumaperekedwa ndi \(P = \rho gh\), pomwe \(\rho\) ndi kuchuluka kwa madzi (pafupifupi \(1000 \text{ kg/m}^3\)) ndipo \(g\) ndi kuthamanga chifukwa cha mphamvu yokoka (pafupifupi \(9.8 \text{ m/s}^2\)).
Kuti tipeze mphamvu yonse yokakamiza, tiyenera kuphatikiza mphamvuyo pamalo oyima a mbale.
Gawo 1: Dziwani ntchito ya kuthamanga kwa magazi.
\[ P(y) = \rho gy \]
Gawo 2: Mphamvu yonse \(F\) ndi gawo lofunikira la nthawi ya kupanikizika kwa dera loyambira \(dA\) kuyambira \(y = 0\) mpaka \(y = 6\).
\[ F = \int_{0}^{6} \rho gy \cdot 4 \, dy \]
Gawo 3: Fewetsani zinthu zosasinthika.
\[ F = 4 \rho g \int_{0}^{6} y \, dy \]
Gawo 4: Dziwani kufunika kwa \(y\).
\[ \int y \, dy = \frac{y^2}{2} \]
Gawo 5: Ikani malire \(0\) ku \(6\).
\[ F = 4 \cdot 1000 \cdot 9.8 \left[ \frac{y^2}{2} \right]_{0}^{6} \]
Gawo 6: Werengani mtengo wokwanira.
\[ F = 4 \cdot 1000 \cdot 9.8 \cdot \frac{6^2}{2} = 4 \cdot 1000 \cdot 9.8 \cdot 18 = 705600 \]
Kotero, mphamvu yonse ya kuthamanga kwa madzi pa mbale ndi \(705600\) Newton.
Mapeto
Kugwiritsa ntchito zinthu zophatikizana m'magwiritsidwe osiyanasiyana kumapereka mphamvu yayikulu yowunikira powerengera kuchuluka kwa zinthu zovuta. M'nkhaniyi, takambirana momwe zinthu zophatikizana zimagwiritsidwira ntchito powerengera dera lomwe lili pansi pa curve, voliyumu ya chinthu cholimba chozungulira, ntchito yochitidwa ndi mphamvu yosinthasintha, ndi kupanikizika kwa hydrostatic. Tikamvetsetsa bwino njira zophatikizira, titha kuthetsa mavuto osiyanasiyana omwe amabuka mu sayansi ndi uinjiniya.
