Mafunso ndi Zitsanzo za Kukambirana za Kugwiritsa Ntchito Zophatikiza Powerengera Malo a Ndege Yathyathyathya
Mu kuphunzitsa masamu, ma integral nthawi zambiri amapezeka mu calculus. Chimodzi mwa ntchito zodziwika bwino za ma integral ndi kuwerengera dera lomwe lili pansi pa curve kapena plane. Nkhaniyi ikambirana mavuto angapo a zitsanzo ndikukambirana za kugwiritsa ntchito ma integral powerengera dera la ndege.
Chiyambi cha Chiphunzitso
Tisanapite ku funso la chitsanzo, tiyeni tiwonenso lingaliro loyambira lowerengera dera lomwe lili pansi pa curve pogwiritsa ntchito ma integrals. Ngati tili ndi ntchito f(x) yomwe imapitilira pa interval [a, b], ndiye kuti dera lomwe lili pansi pa curve y = f(x) kuyambira x = a mpaka x = b ndi:
\[ L = \int_{a}^{b} f(x) \, dx \]
Mwachidule, izi zikutanthauza kuti tikufupikitsa dera la rectangle yopyapyala kwambiri kuyambira x = a mpaka x = b.
Chitsanzo cha Funso 1
Funso
Werengerani dera lomwe lili pansi pa curve y = x² mu interval [1, 3].
Zokambirana
Kuti tiwerengere dera, timagwiritsa ntchito integral:
\[ L = \int_{1}^{3} x^2 \, dx \]
Timayamba ndi kupeza antiderivative ya \( x^2 \). Antiderivative ya \( x^2 \) ndi \( \frac{x^3}{3} \). Kenako integral imakhala:
\[ L = \left[ \frac{x^3}{3} \right]_{1}^{3} \]
Kumbukirani kuti tiyenera kuwunika antiderivative pamalire a integral:
\[ L = \left( \frac{3^3}{3} \right) – \left( \frac{1^3}{3} \right) \]
\[ L = \left( \frac{27}{3} \right) – \left( \frac{1}{3} \right) \]
\[ L = 9 – \frac{1}{3} \]
\[ L = \frac{27}{3} – \frac{1}{3} \]
\[ L = \frac{26}{3} \]
Kotero, dera lomwe lili pansi pa curve y = x² kuchokera x = 1 mpaka x = 3 ndi:
\[ \frac{26}{3} \, \text{area unit} \]
Chitsanzo cha Funso 2
Funso
Dziwani dera la dera lomwe lili ndi malire ndi mzere wozungulira y = x³ ndi mizere x = 1 ndi x = 2.
Zokambirana
Kuti tiwerengere dera, timagwiritsa ntchito integral:
\[ L = \int_{1}^{2} x^3 \, dx \]
Monga mwachizolowezi, timayamba ndi kupeza antiderivative ya \( x^3 \). Antiderivative ya \( x^3 \) ndi \( \frac{x^4}{4} \). Integral imakhala:
\[ L = \left[ \frac{x^4}{4} \right]_{1}^{2} \]
Yesani malire a integral:
\[ L = \left( \frac{2^4}{4} \right) – \left( \frac{1^4}{4} \right) \]
\[ L = \left( \frac{16}{4} \right) – \left( \frac{1}{4} \right) \]
\[ L = 4 – \frac{1}{4} \]
\[ L = \frac{16}{4} – \frac{1}{4} \]
\[ L = \frac{15}{4} \]
Kotero, dera lomwe lili pansi pa curve y = x³ kuchokera x = 1 mpaka x = 2 ndi:
\[ \frac{15}{4} \, \text{area unit} \]
Chitsanzo cha Funso 3
Funso
Dziwani dera la dera lomwe lili ndi malire ndi ma curve y = x² + 1 ndipo y = 2x + 2 pakati pa x = 0 mpaka x = 1.
Zokambirana
Choyamba, tifunika kupeza malo olumikizirana kuti tidziwe malire a kuphatikizana. Yankho la \( x^2 + 1 = 2x + 2 \):
\[ x^2 + 1 = 2x + 2 \]
\[ x^2 – 2x – 1 = 0 \]
Kugwiritsa ntchito quadratic formula:
\[ x = \frac{2 \pm \sqrt{4 + 4}}{2} \]
\[ x = \frac{2 \pm \sqrt{8}}{2} \]
\[ x = \frac{2 \pm 2\sqrt{2}}{2} \]
\[ x = 1 \pm \sqrt{2} \]
Komabe, pa malire apamwamba ndi otsika pakati pa 0 ndi 1, sitifunikira kugwiritsa ntchito yankho la quadratic, koma malire wamba a integral kuyambira 0 mpaka 1. Kenako, werengani dera la top y curve kuchotsa bottom y curve motsatira malire awa:
\[ L = \int_{0}^{1} [(2x + 2) – (x^2 + 1)] \, dx \]
Kusavuta kwa ntchito:
\[ L = \int_{0}^{1} (2x + 2 – x^2 – 1) \, dx \]
\[ L = \int_{0}^{1} (-x^2 + 2x + 1) \, dx \]
Kenako, tikupeza mankhwala oletsa kutupa:
Chotsutsana ndi \( (-x^2) \) ndi \( -\frac{x^3}{3} \),
Choletsa chochokera ku \( (2x) \) ndi \( x^2 \),
Chotsutsana ndi \( (1) \) ndi \( x \).
Ndicholinga choti,
\[ L = \left. \left(-\frac{x^3}{3} + x^2 + x \right) \right|_0^1 \]
Kuwunika kotsatira:
\[ L = \left[ -\frac{1^3}{3} + 1^2 + 1 \right] – \left[ -\frac{0^3}{3} + 0^2 + 0 \right] \]
\[ L = \left[ -\frac{1}{3} + 1 + 1 \right] – \left[ 0 \right] \]
\[ L = -\frac{1}{3} + 2 \]
\[ L = \frac{6}{3} – \frac{1}{3} \]
\[ L = \frac{5}{3} \]
Kotero, dera la dera lomwe lili ndi malire ndi ma curve y = x² + 1 ndipo y = 2x + 2 pa nthawi [0, 1] ndi:
\[ \frac{5}{3} \, \text{area unit} \]
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Kuchokera ku zitsanzo zomwe zili pamwambapa, titha kuwona momwe ma integral angagwiritsidwire ntchito kuwerengera dera lomwe lili pansi pa curve kapena pakati pa ma curve awiri. Pomvetsetsa bwino mfundo zoyambira za ma integral ndi njira zotsutsana ndi ma derivative, kuwerengera madera awa kumakhala koyenera komanso kogwira mtima. Tikukhulupirira kuti nkhaniyi yawonjezera kumvetsetsa kwathu za momwe ma integral amagwiritsidwira ntchito m'dziko lenileni, makamaka pankhani yoyesa dera la malo a ndege.
