Mibvunzo yeMienzaniso neKukurukurirana kweZvishandiso Zvakabatana
Kubatanidzwa ipfungwa huru mukuverenga ine mashandisirwo akawanda muzvikamu zvakasiyana-siyana zvesainzi, zvakaita sefizikisi, economics, biology, uye engineering. MaIntegrals anoshandiswa kuverenga nzvimbo iri pasi pemugero, huwandu hwechinhu chakasimba, basa, kumanikidzwa, nezvimwewo. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yemashandisirwo akabatanidzwa, tichizotevera tsananguro dzakadzama dzemagadzirirwo acho.
1. Kuziva Nzvimbo Iri Pasi Pechinotenderera
Imwe yenzira dzinonyanya kushandiswa dzekubatanidza ndeyekuverenga nzvimbo iri pasi pecurve yebasa mukati menguva yakatarwa. Ngatitii tinoda kuwana nzvimbo yenzvimbo yakaganhurirwa necurve \(y = x^2\) uye \(x\) axis kubva \(x = 0\) kusvika \(x = 2\).
Muenzaniso wematambudziko:
Sarudza nzvimbo iri pasi pekona \(y = x^2\) kubva \(x = 0\) kusvika \(x = 2\).
Kukurukurirana:
Kuti tiwane nzvimbo iri pasi pe curve \(y = x^2\) kubva \(x = 0\) kusvika \(x = 2\), tinofanira kuverenga definite integral yebasa:
\[ \int_{0}^{2} x^2 \, dx \]
Danho 1: Sarudza chinhu chakakosha che \(x^2\).
Cherechedza kuti chinhu chakakosha che \(x^2\) ndeichi:
\[ \int x^2 \, dx = \frac{x^3}{3} + C \]
Danho rechipiri: Isa muganho we integral \(0\) ku \(2\).
\[ \int_{0}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{2} \]
Danho rechitanhatu: Verenga kukosha kwemuganho.
\[ \kuruboshwe. \frac{x^3}{3} \kurudyi|_{0}^{2} = \frac{2^3}{3} – \frac{0^3}{3} = \frac{8}{3} – 0 = \frac{8}{3} \]
Saka, nzvimbo iri pasi pe curve \(y = x^2\) kubva \(x = 0\) kusvika \(x = 2\) ndiyo \( \frac{8}{3} \) area units.
2. Kuverenga Vhoriyamu Yezvinhu Zvinotenderera
MaIntegrals anoshandiswawo kuverenga huwandu hwezvinhu zvakasimba zvekuchinja. Kana nzvimbo ikatenderedzwa nepakati pe \(x\) axis, huwandu hwechinhu hunogona kuwanikwa uchishandisa nzira yedhisiki kana nzira yeringi.
Muenzaniso wematambudziko:
Verenga huwandu hwechinhu chinogadzirwa kana nzvimbo yakaganhurirwa ne curve \(y = \sqrt{x}\) uye mutsetse \(x = 4\) zvatenderedzwa kutenderedza \(x\) axis.
Kukurukurirana:
Kuti tiwane vhoriyamu yechinhu chakasimba chinotenderera, tinogona kushandisa nzira yedhisiki. Vhoriyamu \(V\) yechinhu chakasimba chinobuda inogona kuratidzwa seizvi:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
Kupi \(f(x) = \sqrt{x}\), \(a = 0\), uye \(b = 4\).
Danho 1: Gadzira chikamu chevhoriyamu.
\[ V = \pi \int_{0}^{4} (\sqrt{x})^2 \, dx \]
Danho rechipiri: Rerutsa basa riri muchikamu chekubatanidza.
\[ V = \pi \int_{0}^{4} x \, dx \]
Danho rechitatu: Sarudza chinhu chakakosha che \(x\).
\[ \int x \, dx = \frac{x^2}{2} + C \]
Danho rechishanu: Isa miganhu \(0\) ku \(4\).
\[ V = \pi \kuruboshwe[ \frac{x^2}{2} \kurudyi]_{0}^{4} \]
Danho rechitanhatu: Verenga kukosha kwemuganho.
\[ \kuruboshwe. \frac{x^2}{2} \kurudyi|_{0}^{4} = \pi \kuruboshwe( \frac{4^2}{2} – \frac{0^2}{2} \kurudyi) = \pi \kuruboshwe( \frac{16}{2} \kurudyi) = 8\pi \]
Saka, vhoriyamu yechinhu chinobuda i \(8\pi\) mayuniti evhoriyamu.
3. Kuverenga Basa Rinoitwa Nesimba Rinochinja-chinja
Mashandisirwo akabatanidzwa anowanikwawo mufizikisi, imwe yacho ndeyekuverenga basa rinoitwa nesimba rinoshanduka kana chinhu chichifamba kubva pane imwe nzvimbo kuenda kune imwe.
Muenzaniso wematambudziko:
Simba \(F(x) = 3x^2\) Newton inoshanda pane chinhu chinofamba kubva pa \(x = 1\) mita kuenda pa \(x = 3\) mamita. Verenga basa rinoitwa nesimba racho.
Kukurukurirana:
Basa \(W\) rinoitwa nesimba \(F(x)\) rinogona kuwanikwa nekuverenga chikamu che \(F(x)\) pamusoro pekuchinja kubva \(a\) kuenda \(b\):
\[ W = \int_{a}^{b} F(x) \, dx \]
Kupi \(a = 1\), \(b = 3\), uye \(F(x) = 3x^2\).
Danho 1: Gadzira chikamu chakakosha chebasa racho.
\[ W = \int_{1}^{3} 3x^2 \, dx \]
Danho rechipiri: Sarudza chinhu chakakosha che \(3x^2\).
\[ \int 3x^2 \, dx = 3 \kuruboshwe( \frac{x^3}{3} \kurudyi) = x^3 + C \]
Danho rechishanu: Isa miganhu \(1\) ku \(3\).
\[ W = \kuruboshwe[ x^3 \kurudyi]_{1}^{3} \]
Danho rechitanhatu: Verenga kukosha kwemuganho.
\[ W = \kuruboshwe. x^3 \kurudyi|_{1}^{3} = 3^3 – 1^3 = 27 – 1 = 26 \]
Saka, basa rinoitwa nemauto i \(26\) joules.
4. Kuziva Kumanikidzwa Kwemvura
Mufizikisi, ma integrals anoshandiswawo kuverenga hydrostatic pressure pamusoro pemvura yakanyudzwa mumvura.
Muenzaniso wematambudziko:
Ndiro yakamira yakareba mamita matanhatu uye yakafara mamita mana inonyudzwa mumvura, pamusoro payo pari pamusoro pemvura. Verenga simba rose rekumanikidzwa kwemvura pandiro.
Kukurukurirana:
Kumanikidzwa pakadzika \(h\) mumvura kunopiwa ne \(P = \rho gh\), apo \(\rho\) iri huwandu hwemvura (inenge \(1000 \text{ kg/m}^3\)) uye \(g\) iri kukurumidza kunokonzerwa negiravhiti (inenge \(9.8 \text{ m/s}^2\)).
Kuti tiwane simba rekumanikidza rose, tinofanira kubatanidza kumanikidzwa pamusoro penzvimbo yakatwasuka yeplate.
Danho 1: Sarudza mashandiro ekumanikidza.
\[ P(y) = \rho gy \]
Danho rechipiri: Simba rose \(F\) ndiro chinhu chinobatanidza kumanikidzwa kunoenderana nenzvimbo yekutanga \(dA\) kubva \(y = 0\) kusvika \(y = 6\).
\[ F = \int_{0}^{6} \rho gy \cdot 4 \, dy \]
Danho rechitatu: Rerutsa zvirambidzo.
\[ F = 4 \rho g \int_{0}^{6} y \, dy \]
Danho rechina: Sarudza chinhu chakakosha che \(y\).
\[ \int y \, dy = \frac{y^2}{2} \]
Danho rechishanu: Isa miganhu \(0\) ku \(6\).
\[ F = 4 \cdot 1000 \cdot 9.8 \left[ \frac{y^2}{2} \right]_{0}^{6} \]
Danho rechitanhatu: Verenga kukosha kwemuganho.
\[ F = 4 \cdot 1000 \cdot 9.8 \cdot \frac{6^2}{2} = 4 \cdot 1000 \cdot 9.8 \cdot 18 = 705600 \]
Saka, simba rose rekumanikidzwa kwemvura paplate ndi \(705600\) Newton.
Mhedziso
Kushandiswa kwezvinobatanidza mumabasa akasiyana-siyana kunopa simba guru rekuongorora pakuverenga huwandu hwakaoma hwemuviri. Muchinyorwa chino, takurukura kuti zvinobatanidza zvinoshandiswa sei pakuverenga nzvimbo iri pasi pemugero, vhoriyamu yechinhu chakasimba chekuchinja, basa rinoitwa nesimba rinoshanduka, uye kumanikidzwa kwemvura. Nekunzwisisa kwakanaka kwehunyanzvi hwekubatanidza, tinogona kugadzirisa matambudziko akasiyana-siyana anoitika musainzi neinjiniya.
