Ajụjụ Ihe Nlereanya na Mkparịta ụka nke Ngwa Ndị Dị n'Otu
Njikọta bụ echiche dị mkpa na mgbakọ na mwepụ nke nwere ọtụtụ ojiji na ngalaba sayensị dị iche iche, dịka fizikisi, akụnụba, bayoloji, na injinia. A na-eji njikọta agbakọ mpaghara dị n'okpuru mgbagọ, olu nke ihe siri ike, ọrụ, nrụgide, na ihe ndị ọzọ. N'isiokwu a, anyị ga-atụle ọtụtụ ihe atụ nke ngwa dị n'ime, wee nye nkọwa zuru ezu banyere otu esi edozi ha.
1. Ịchọpụta Mpaghara Dị n'okpuru Mgbidi ahụ
Otu n'ime ojiji nke ihe ndị a na-ejikarị eme ihe bụ ịgbakọ mpaghara dị n'okpuru usoro ọrụ n'ime oge enyere. Ka e were ya na anyị chọrọ ịchọta mpaghara nke mpaghara ahụ nke usoro \(y = x^2\) na axis \(x\) sitere na \(x = 0\) ruo \(x = 2\).
Ihe atụ nke nsogbu:
Chọpụta mpaghara dị n'okpuru mgbagọ ahụ \(y = x^2\) site na \(x = 0\) ruo \(x = 2\).
Azịza:
Iji chọta mpaghara dị n'okpuru mgbagọ ahụ \(y = x^2\) site na \(x = 0\) ruo \(x = 2\), anyị kwesịrị ịgbakọ ihe mejupụtara ọrụ ahụ:
\[ \int_{0}^{2} x^2 \, dx \]
Nzọụkwụ 1: Chọpụta ihe dị n'ime \(x^2\).
Rịba ama na ihe mejupụtara \(x^2\) bụ:
\[ \int x^2 \, dx = \frac{x^3}{3} + C \]
Nzọụkwụ nke 2: Tinye oke integral \(0\) na \(2\).
\[ \int_{0}^{2} x^2 \, dx = \ekpe[ \frac{x^3}{3} \nri]_{0}^{2} \]
Nzọụkwụ nke 3: Gbakọọ uru oke.
\[ \aka ekpe. \frac{x^3}{3} \right|_{0}^{2} = \frac{2^3}{3} – \frac{0^3}{3} = \frac{8}{3} – 0 = \frac{8}{3} \]
Ya mere, mpaghara dị n'okpuru mgbagọ \(y = x^2\) site na \(x = 0\) ruo \(x = 2\) bụ nkeji mpaghara \( \frac{8}{3} \).
2. Ịgbakọ olu nke ihe ndị na-agbagharị
A na-ejikwa Integrals agbakọ olu nke ihe siri ike nke mgbanwe. Ọ bụrụ na a na-atụgharị mpaghara gburugburu axis \(x\), a pụrụ ịchọta olu nke ihe ahụ site na iji usoro diski ma ọ bụ usoro mgbanaka.
Ihe atụ nke nsogbu:
Gbakọọ olu nke ihe e mepụtara mgbe mpaghara nke usoro ahụ gbara ya gburugburu na ahịrị \(y = \sqrt{x}\) na-atụgharị ahịrị \(x = 4\) gburugburu axis \(x\).
Azịza:
Iji chọta olu nke ihe siri ike nke mgbanwe, anyị nwere ike iji usoro diski ahụ. Enwere ike ịkọwa olu \(V\) nke ihe siri ike a na-enweta dị ka:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
Ebe \(f(x) = \sqrt{x}\), \(a = 0\), na \(b = 4\).
Nzọụkwụ 1: Mepụta ihe dị mkpa nke olu.
\[ V = \pi \int_{0}^{4} (\sqrt{x})^2 \, dx \]
Nzọụkwụ nke 2: Mee ka ọrụ dị na integral dị mfe.
\[ V = \pi \int_{0}^{4} x \, dx \]
Nzọụkwụ nke 3: Chọpụta ihe dị n'ime \(x\).
\[ \int x \, dx = \frac{x^2}{2} + C \]
Nzọụkwụ nke 4: Tinye oke \(0\) gaa na \(4\).
\[ V = \pi \ekpe[ \frac{x^2}{2} \nri]_{0}^{4} \]
Nzọụkwụ nke 5: Gbakọọ uru oke.
\[ \left. \frac{x^2}{2} \right|_{0}^{4} = \pi \left( \frac{4^2}{2} – \frac{0^2}{2} \right) = \pi \left( \frac{16}{2} \right) = 8\pi \]
Ya mere, olu nke ihe si na ya pụta bụ nkeji olu \(8\pi\).
3. Ịgbakọ Ọrụ nke Ike Na-agbanwe Agbanwe Na-arụ
A na-ahụkwa ngwa ndị e ji eme ihe n'ime fiziki, otu n'ime ha bụ ịgbakọ ọrụ ike mgbanwe rụrụ mgbe ihe si n'otu ebe gaa na nke ọzọ.
Ihe atụ nke nsogbu:
Ike \(F(x) = 3x^2\) Newton na-arụ ọrụ na obere ihe na-esi na mita \(x = 1\) gaa na mita \(x = 3\). Gbakọọ ọrụ ike ahụ rụrụ.
Azịza:
Enwere ike ịchọta ọrụ \(W\) nke ike \(F(x)\) mere site na ịgbakọ ihe mejupụtara \(F(x)\) n'elu mgbanwe site na \(a\) gaa na \(b\):
\[ W = \int_{a}^{b} F(x) \, dx \]
Ebe \(a = 1\), \(b = 3\), na \(F(x) = 3x^2\).
Nzọụkwụ 1: Mepụta ihe dị mkpa nke ọrụ ahụ.
\[ W = \int_{1}^{3} 3x^2 \, dx \]
Nzọụkwụ nke 2: Chọpụta ihe dị n'ime \(3x^2\).
\[ \int 3x^2 \, dx = 3 \ekpe( \frac{x^3}{3} \nri) = x^3 + C \]
Nzọụkwụ nke 3: Tinye oke \(1\) gaa na \(3\).
\[ W = \ekpe[ x^3 \nri]_{1}^{3} \]
Nzọụkwụ nke 4: Gbakọọ uru oke.
\[ W = \aka ekpe. x^3 \aka nri|_{1}^{3} = 3^3 – 1^3 = 27 – 1 = 26 \]
Ya mere, ọrụ ndị ike rụrụ bụ joules.
4. Ịchọpụta Nrụgide Mmiri
Na fizikisi, a na-ejikwa ihe ndị mejupụtara ihe agbakọọ nrụgide hydrostatic n'elu mmiri mmiri.
Ihe atụ nke nsogbu:
A na-emikpu efere kwụ ọtọ nke dị mita isii n'ịdị elu na mita anọ n'obosara n'ime mmiri, elu ya dịkwa n'elu mmiri. Gbakọọ ike niile nke nrụgide mmiri dị na efere ahụ.
Azịza:
A na-enye nrụgide dị n'omimi \(h\) n'ime mmiri site na \(P = \rho gh\), ebe \(\rho\) bụ njupụta nke mmiri (ihe dị ka \(1000 \text{ kg/m}^3\)) na \(g\) bụ ọsọ ọsọ n'ihi ike ndọda (ihe dị ka \(9.8 \text{ m/s}^2\)).
Maka ike nrụgide zuru oke, anyị ga-ejikọta nrụgide ahụ n'elu mpaghara kwụ ọtọ nke efere ahụ.
Nzọụkwụ 1: Chọpụta ọrụ nrụgide.
\[ P(y) = \rho gy \]
Nzọụkwụ nke 2: Ike niile \(F\) bụ ihe dị mkpa nke nrụgide ugboro mpaghara mbụ \(dA\) site na \(y = 0\) ruo \(y = 6\).
\[ F = \int_{0}^{6} \rho gy \cdot 4 \, dy \]
Nzọụkwụ nke 3: Mee ka ihe ndị na-agbanwe agbanwe dị mfe.
\[ F = 4 \rho g \int_{0}^{6} y \, dy \]
Nzọụkwụ nke 4: Chọpụta ihe dị n'ime \(y\).
\[ \int y \, dy = \frac{y^2}{2} \]
Nzọụkwụ nke 5: Tinye oke \(0\) gaa na \(6\).
\[ F = 4 \cdot 1000 \cdot 9.8 \ekpe[ \frac{y^2}{2} \nri]_{0}^{6} \]
Nzọụkwụ nke 6: Gbakọọ uru oke.
\[ F = 4 \cdot 1000 \cdot 9.8 \cdot \frac{6^2}{2} = 4 \cdot 1000 \cdot 9.8 \cdot 18 = 705600 \]
Ya mere, ike zuru oke nke nrụgide mmiri dị na efere ahụ bụ \(705600\) Newton.
Mmechi
Ojiji nke integrals n'ọtụtụ ngwa na-enye ike nyocha dị ukwuu maka ịgbakọ ọnụọgụ anụ ahụ dị mgbagwoju anya. N'isiokwu a, anyị atụleela otu esi etinye integrals iji gbakọọ mpaghara dị n'okpuru usoro, olu nke mgbanwe siri ike, ọrụ nke ike mgbanwe na-arụ, na nrụgide hydrostatic. Site na nghọta dị mma nke usoro njikọta, anyị nwere ike idozi ọtụtụ nsogbu bara uru nke na-ebilite na sayensị na injinia.
