Cov Lus Nug Piv Txwv thiab Kev Sib Tham Txog Kev Siv Ua Ib Ke
Kev sib xyaw ua ke yog ib lub tswv yim tseem ceeb hauv kev xam lej uas muaj ntau yam kev siv hauv ntau qhov chaw ntawm kev tshawb fawb, xws li physics, economics, biology, thiab engineering. Integrals yog siv los xam thaj tsam hauv qab ib daim duab nkhaus, qhov ntim ntawm ib yam khoom khov, kev ua haujlwm, siab, thiab ntau ntxiv. Hauv tsab xov xwm no, peb yuav tham txog ntau qhov piv txwv ntawm kev siv integral, ua raws li cov lus piav qhia ntxaws txog yuav ua li cas daws lawv.
1. Txheeb Xyuas Thaj Chaw Hauv Qab Qhov nkhaus
Ib qho ntawm cov ntawv thov feem ntau ntawm cov integrals yog xam thaj tsam hauv qab qhov nkhaus ntawm ib qho kev ua haujlwm dhau ib lub sijhawm muab. Xav tias peb xav nrhiav thaj tsam ntawm thaj tsam uas txwv los ntawm qhov nkhaus \(y = x^2\) thiab qhov axis \(x\) los ntawm \(x = 0\) mus rau \(x = 2\).
Piv txwv teeb meem:
Txheeb xyuas thaj tsam hauv qab tus nkhaus \(y = x^2\) txij ntawm \(x = 0\) mus txog \(x = 2\).
Kev Sib Tham:
Yuav nrhiav tau thaj tsam hauv qab tus nkhaus \(y = x^2\) los ntawm \(x = 0\) mus rau \(x = 2\), peb yuav tsum xam qhov integral tseeb ntawm qhov kev ua haujlwm:
\[ \int_{0}^{2} x^2 \, dx \]
Kauj Ruam 1: Txheeb xyuas qhov sib npaug ntawm \(x^2\).
Nco ntsoov tias qhov sib xyaw ua ke ntawm \(x^2\) yog:
\[ \int x^2 \, dx = \frac{x^3}{3} + C \]
Kauj Ruam 2: Siv qhov txwv ntawm qhov sib xyaw ua ke \(0\) rau \(2\).
\[ \int_{0}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{2} \]
Kauj Ruam 3: Xam tus nqi txwv.
\[ \left. \frac{x^3}{3} \right|_{0}^{2} = \frac{2^3}{3} – \frac{0^3}{3} = \frac{8}{3} – 0 = \frac{8}{3} \]
Yog li, thaj chaw hauv qab nkhaus \(y = x^2\) los ntawm \(x = 0\) mus rau \(x = 2\) yog \(\frac{8}{3}\) cov chav nyob.
2. Xam Qhov Ntim ntawm Cov Khoom Tig
Cov integrals kuj siv los xam qhov ntim ntawm cov khoom khov uas tig rov qab. Yog tias ib cheeb tsam tig ib ncig ntawm lub axis \(x\), qhov ntim ntawm cov khoom tuaj yeem nrhiav tau siv txoj kev disk lossis txoj kev nplhaib.
Piv txwv teeb meem:
Xam qhov ntim ntawm cov khoom tsim tawm thaum thaj tsam uas ciam teb los ntawm qhov nkhaus \(y = \sqrt{x}\) thiab txoj kab \(x = 4\) tig ib ncig ntawm \(x\) axis.
Kev Sib Tham:
Yuav nrhiav qhov ntim ntawm cov khoom khov uas tig rov qab, peb siv tau txoj kev disk. Qhov ntim \(V\) ntawm cov khoom khov uas tau los ntawm qhov no tuaj yeem qhia tau li no:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
Qhov twg f(x) = x\), a = 0\, thiab b = 4\.
Kauj Ruam 1: Tsim lub ntim integral.
\[ V = \pi \int_{0}^{4} (\sqrt{x})^2 \, dx \]
Kauj Ruam 2: Ua kom yooj yim rau txoj haujlwm hauv qhov sib xyaw ua ke.
\[ V = \pi \int_{0}^{4} x \, dx \]
Kauj Ruam 3: Txheeb xyuas qhov sib npaug ntawm \(x\).
\[ \int x\, dx = \frac{x^2}{2} + C\]
Kauj Ruam 4: Siv cov kev txwv \(0\) rau \(4\).
\[ V = \pi \left[ \frac{x^2}{2} \right]_{0}^{4} \]
Kauj Ruam 5: Xam tus nqi txwv.
\[ \left. \frac{x^2}{2} \right|_{0}^{4} = \pi \left( \frac{4^2}{2} – \frac{0^2}{2} \right) = \pi \left( \frac{16}{2} \right) = 8\pi \]
Yog li, qhov ntim ntawm cov khoom tshwm sim yog \(8\pi\) units ntawm ntim.
3. Xam Cov Haujlwm Uas Ua Los Ntawm Lub Zog Hloov Pauv
Cov ntawv thov sib xyaw ua ke kuj pom muaj nyob rau hauv physics, ib qho ntawm cov no yog xam cov haujlwm ua los ntawm lub zog hloov pauv thaum ib yam khoom txav ntawm ib qho mus rau lwm qhov.
Piv txwv teeb meem:
Ib lub zog \(F(x) = 3x^2\) Newton ua rau ib qho khoom me me txav ntawm \(x = 1\) 'meter' mus rau \(x = 3\) 'meter'. Xam qhov haujlwm uas lub zog ua tiav.
Kev Sib Tham:
Txoj haujlwm \(W\) ua los ntawm lub zog \(F(x)\) tuaj yeem nrhiav tau los ntawm kev xam qhov sib xyaw ntawm \(F(x)\) hla qhov kev hloov chaw ntawm \(a\) mus rau \(b\):
\[ W = \int_{a}^{b} F(x) \, dx \]
Qhov twg \(a = 1\), \(b = 3\), thiab \(F(x) = 3x^2\).
Kauj Ruam 1: Tsim qhov sib xyaw ua ke ntawm txoj haujlwm.
\[ W = \int_{1}^{3} 3x^2 \, dx \]
Kauj Ruam 2: Txheeb xyuas qhov integral ntawm \(3x^2\).
\[ \int 3x^2 \, dx = 3 \left( \frac{x^3}{3} \right) = x^3 + C \]
Kauj Ruam 3: Siv cov kev txwv \(1\) rau \(3\).
\[ W = \left[ x^3 \right]_{1}^{3} \]
Kauj Ruam 4: Xam tus nqi txwv.
\[ W = \left. x^3 \right|_{1}^{3} = 3^3 – 1^3 = 27 – 1 = 26 \]
Yog li, txoj haujlwm ua los ntawm lub zog yog ∙ (26) joules.
4. Kev Txheeb Xyuas Qhov Siab Hydrostatic
Hauv physics, integrals kuj tseem siv los xam qhov hydrostatic siab ntawm qhov chaw uas muab tso rau hauv cov kua.
Piv txwv teeb meem:
Muaj ib lub phaj ntsug uas siab 6 meters thiab dav 4 meters raug muab tso rau hauv dej thiab sab saud nyob saum dej. Xam tag nrho lub zog ntawm dej siab rau ntawm lub phaj.
Kev Sib Tham:
Lub siab ntawm qhov tob ∑(h) hauv dej yog muab los ntawm ∑(P = ∑(rh)gh), qhov twg ∑(rh) yog qhov ceev ntawm dej (kwv yees li ∑(1000 kg/m3)) thiab ∑(g) yog qhov nrawm vim lub ntiajteb txawj nqus (kwv yees li ∑(9.8 m/s)^2).
Rau tag nrho cov zog siab, peb yuav tsum sib koom ua ke lub zog siab hla thaj tsam ntsug ntawm lub phaj.
Kauj Ruam 1: Txheeb xyuas qhov ua haujlwm siab.
\[ P(y) = \rho gy \]
Kauj Ruam 2: Tag nrho lub zog \(F\) yog qhov sib xyaw ntawm lub zog siab zaug rau thaj chaw pib \(dA\) txij \(y = 0\) mus txog \(y = 6\).
\[ F = \int_{0}^{6} \rho gy \cdot 4 \, dy \]
Kauj Ruam 3: Ua kom yooj yim cov constants.
\[ F = 4 \rho g \int_{0}^{6} y \, dy \]
Kauj Ruam 4: Txheeb xyuas qhov integral ntawm \(y\).
\[ \int y \, dy = \frac{y^2}{2} \]
Kauj Ruam 5: Siv cov kev txwv \(0\) rau \(6\).
\[ F = 4 \cdot 1000 \cdot 9.8 \left[ \frac{y^2}{2} \right]_{0}^{6} \]
Kauj Ruam 6: Xam tus nqi txwv.
\[ F = 4 \cdot 1000 \cdot 9.8 \cdot \frac{6^2}{2} = 4 \cdot 1000 \cdot 9.8 \cdot 18 = 705600 \]
Yog li, tag nrho lub zog ntawm dej siab rau ntawm lub phaj yog \(705600\) Newton.
Xaus
Kev siv cov integrals hauv ntau daim ntawv thov muab lub zog loj heev rau kev xam cov khoom sib xyaw ua ke. Hauv tsab xov xwm no, peb tau tham txog yuav ua li cas siv cov integrals los xam thaj tsam hauv qab ib qho nkhaus, qhov ntim ntawm cov khoom khov kho ntawm kev hloov pauv, txoj haujlwm ua los ntawm lub zog hloov pauv, thiab lub zog hydrostatic. Nrog kev nkag siab zoo txog cov txheej txheem kev sib xyaw ua ke, peb tuaj yeem daws ntau yam teeb meem uas tshwm sim hauv kev tshawb fawb thiab kev tsim kho.
