Imibuzo Eyisibonelo Exoxa Ngokusetshenziswa Kwezinto Ezihlanganisiwe Emkhakheni Wezomnotho Nebhizinisi
I-Pendahuluan
Ama-Integral angumqondo oyinhloko ekubaleni futhi anezinhlelo zokusebenza eziningi emikhakheni ehlukahlukene, okuhlanganisa ezomnotho kanye nebhizinisi. Kulesi simo, ama-integral avame ukusetshenziswa ukuhlaziya inzuzo iyonke, izindleko, imali engenayo, kanye nemisebenzi yokusetshenziswa kanye nokukhiqiza. Ukuqonda ukusetshenziswa kwama-integral kwezomnotho kanye nebhizinisi akusizi nje kuphela ukuxazulula izinkinga zobuchwepheshe kodwa futhi kunikeza ukuqonda okujulile mayelana nokuguquguquka kwemakethe, ukwenza izinqumo, kanye nokuhlela amasu.
Izicelo Ezihlanganisiwe Kwezomnotho Nasebhizinisini
1. Bala Imali Engenayo Yonke
Ukuze sibale imali engenayo iyonke, sivame ukudinga ukuhlanganisa imali engenayo encane etholwe ngokuthengiswa kwamayunithi ngamanye omkhiqizo. Uma intengo yomkhiqizo ihluka kuye ngenani elithengisiwe, khona-ke umsebenzi wentengo-ubuningi kumele uhlanganiswe ukuze kunqunywe imali engenayo iyonke.
Isibonelo sezinkinga:
Ake sithi intengo \( p \) yempahla incike enanini lomkhiqizo \( q \) othengisiwe, okunikezwa umsebenzi olandelayo:
\[p(q) = 100 – 2q \]
Bala imali engenayo iyonke uma kuthengiswa amayunithi ayi-10 ezimpahla.
Isixazululo:
Imali engenayo iyonke \(R \) iyinhlanganisela yentengo ngaphezu kwenani elisukela kumayunithi angu-0 kuya kwangu-\(Q \).
\[ R = \int_{0}^{Q} p(q) \, dq \]
Nge-\( p(q) = 100 – 2q \) kanye ne-\( Q = 10 \):
\[ R = \int_{0}^{10} (100 – 2q) \, dq \]
Ngakho-ke, sibala i-integral:
\[ R = \kwesobunxele[ 100q – q^2 \kwesokudla]_{0}^{10} \]
Hlola imikhawulo ye-integral:
\[ R = \kwesobunxele( 100 \cdot 10 – 10^2 \kwesokudla) – \kwesobunxele( 100 \cdot 0 – 0^2 \kwesokudla) \]
\[ = 1000 – 100 \]
\[ = 900 \]
Ngakho-ke, imali engenayo iyonke uma kuthengiswa amayunithi ayi-10 ezimpahla ingu-900.
2. Bala Izindleko Eziphelele
Ukusetshenziswa kwezinto ezihlanganisiwe ekubaleni izindleko zokukhiqiza eziphelele kuwusizo kakhulu, ikakhulukazi lapho izindleko ezingaphansi kwemingcele zingaguquguquki futhi zincike enanini elikhiqizwayo. Izindleko ezingaphansi kwemingcele zingachazwa njenge-derivative yezindleko eziphelele, futhi ukuze sithole izindleko eziphelele sidinga ukuzihlanganisa.
Isibonelo sezinkinga:
Uma izindleko ezingaphansi \(MC \) zokukhiqiza \(q \) amayunithi empahla zinikezwa ngu:
\[ MC(q) = 50 + 3q^2 \]
Bala izindleko eziphelele uma kukhiqizwa amayunithi ama-5 ezimpahla uma kucatshangelwa izindleko ezihleliwe \(C \) ezingama-200.
Isixazululo:
Izindleko eziphelele \(TC \) ziyinhlanganisela yezindleko ezisemaphethelweni kanye nezindleko ezihleliwe:
\[ TC = \int_{0}^{Q} MC(q) \, dq + C \]
Nge-\( MC(q) = 50 + 3q^2 \) kanye ne-\( Q = 5 \):
\[ TC = \int_{0}^{5} (50 + 3q^2) \, dq + 200 \]
Sibala i-integral:
\[ TC = \kwesobunxele[ 50q + q^3 \kwesokudla]_{0}^{5} + 200 \]
Hlola imikhawulo ye-integral:
\[ TC = \kwesobunxele( 50 \cdot 5 + 5^3 \kwesokudla) – \kwesobunxele( 50 \cdot 0 + 0^3 \kwesokudla) + 200 \]
\[ = \kwesobunxele( 250 + 125 \kwesokudla) + 200 \]
\[ = 375 + 200 \]
\[ = 575 \]
Ngakho-ke, izindleko eziphelele zokukhiqiza amayunithi ama-5 ezimpahla zingama-575.
3. Ukubala Ukusetshenziswa Kwezinsizakusebenza
Ama-Integrals asetshenziswa futhi ukubala ukusetshenziswa noma ukusetshenziswa okuphelele kwesisetshenziswa esikhathini esithile. Lokhu kubaluleke kakhulu ezimweni zebhizinisi ezihilela izinsiza ezifana namandla, izinto zokwakha, noma abantu.
Isibonelo sezinkinga:
Izinga lokusetshenziswa kwamandla nsuku zonke \(E \) efektri lilandela umsebenzi olandelayo we-exponential:
\[ E(t) = 10e^{0.1t} \]
Bala inani lamandla asetshenziswayo izinsuku eziyi-10.
Isixazululo:
Ukusetshenziswa kwamandla okuphelele \( C \) phakathi nesikhathi [0, T] kuyingxenye ebalulekile yalawo mazinga okusetshenziswa kwamandla:
\[ C = \int_{0}^{T} E(t) \, dt \]
Nge-\( E(t) = 10e^{0.1t} \) kanye ne-\( T = 10 \):
\[ C = \int_{0}^{10} 10e^{0.1t} \, dt \]
Ukuze sibale i-integral, singasebenzisa indlela yokufaka esikhundleni:
Ake \( u = 0.1t \), bese kuba \( du = 0.1 \, dt \), noma \( dt = \frac{du}{0.1} \),
\[ C = \int_{0}^{1} 10e^{u} \frac{du}{0.1} \]
\[ = 100 \int_{0}^{1} e^{u} \, du \]
\[ = 100 \kwesobunxele[ e^{u} \kwesokudla]_{0}^{1} \]
Hlola imikhawulo ye-integral:
\[ C = 100 \kwesobunxele( e^{1} – e^{0} \kwesokudla) \]
\[ = 100 \kwesobunxele( e – 1 \kwesokudla) \]
Nge \( e \approx 2.718 \):
\[ C \cishe kube yi-100 (2.718 - 1) \]
\[ = 100 \izikhathi 1.718 \]
\[ = 171.8 \]
Ngakho-ke, ukusetshenziswa kwamandla okuphelele kwezinsuku eziyi-10 kungamayunithi angu-171.8 wamandla.
Isiphetho
Umqondo wezinto ezihlanganisiwe ubalulekile kwezomnotho nasebhizinisini, njengoba uvumela abahlaziyi kanye nabenzi bezinqumo ukubala nokubikezela izinto ezibalulekile njengemali engenayo, izindleko, kanye nokusetshenziswa. Ukuqonda ukuthi zisetshenziswa kanjani izinto ezihlanganisiwe kulezi zimo ezahlukahlukene kungakunikeza inzuzo yokuncintisana kanye nokuqonda okungcono ngokusebenza kwebhizinisi. Ngethemba ukuthi lezi zinkinga zezibonelo zizokusiza uqonde ukusetshenziswa okusebenzayo kwezinto ezihlanganisiwe kwezomnotho nasebhizinisini.
