Tambayoyi da Tattaunawa game da Amfani da Haɗaka wajen Ƙididdige Yankin Jirgin Sama Mai Faɗi
A cikin koyarwar lissafi, ana samun haɗaka sau da yawa a cikin kalkuleta. Ɗaya daga cikin shahararrun aikace-aikacen haɗaka shine ƙididdige yankin da ke ƙarƙashin lanƙwasa ko jirgin sama. Wannan labarin zai tattauna misalai da yawa na matsaloli kuma ya tattauna amfani da haɗaka don ƙididdige yankin jirgin sama.
Gabatarwa ga Ka'ida
Kafin mu ci gaba zuwa ga matsalar misali, bari mu sake duba manufar asali ta ƙididdige yankin da ke ƙarƙashin lanƙwasa ta amfani da haɗin gwiwa. Idan muna da aikin f(x) wanda yake ci gaba a kan tazara [a, b], to yankin da ke ƙarƙashin lanƙwasa y = f(x) daga x = a zuwa x = b shine:
\[ L = \int_{a}^{b} f(x) \, dx \]
A fannin lissafi, wannan yana nufin muna taƙaita yankin murabba'i mai siriri daga x = a zuwa x = b.
Misali Tambaya ta 1
Sol
Lissafa yankin da ke ƙarƙashin lanƙwasa y = x² a cikin tazara [1, 3].
Tattaunawa
Don ƙididdige yankin, muna amfani da haɗin:
\[ L = \int_{1}^{3} x^2 \, dx \]
Za mu fara da nemo antiderivative na \( x^2 \). Antiderivative na \( x^2 \) shine \( \frac{x^3}{3} \). Sannan integral ya zama:
\[ L = \left[ \frac{x^3}{3} \right]_{1}^{3} \]
Ka tuna cewa dole ne mu kimanta antiderivative akan iyakokin integral:
\[ L = \left( \frac{3^3}{3} \right) – \left( \frac{1^3}{3} \right) \]
\[ L = \left( \frac{27}{3} \right) – \left( \frac{1}{3} \right) \]
\[ L = 9 – \frac{1}{3} \]
\[ L = \frac{27}{3} – \frac{1}{3} \]
\[ L = \frac{26}{3} \]
Don haka, yankin da ke ƙarƙashin lanƙwasa y = x² daga x = 1 zuwa x = 3 shine:
\[ \frac{26}{3} \, \text{naúrar yanki} \]
Misali Tambaya ta 2
Sol
Kayyade yankin yankin da aka iyakance da lanƙwasa y = x³ da layukan x = 1 da x = 2.
Tattaunawa
Don ƙididdige yankin, muna amfani da haɗin:
\[ L = \int_{1}^{2} x^3 \, dx \]
Kamar yadda aka saba, muna farawa da nemo antiderivative na \( x^3 \). Antiderivative na \( x^3 \) shine \( \frac{x^4}{4} \). Haɗin ya zama:
\[ L = \left[ \frac{x^4}{4} \right]_{1}^{2} \]
Kimanta iyakokin haɗin:
\[ L = \left( \frac{2^4}{4} \right) – \left( \frac{1^4}{4} \right) \]
\[ L = \left( \frac{16}{4} \right) – \left( \frac{1}{4} \right) \]
\[ L = 4 – \frac{1}{4} \]
\[ L = \frac{16}{4} – \frac{1}{4} \]
\[ L = \frac{15}{4} \]
Don haka, yankin da ke ƙarƙashin lanƙwasa y = x³ daga x = 1 zuwa x = 2 shine:
\[ \frac{15}{4} \, \text{naúrar yanki} \]
Misali Tambaya ta 3
Sol
Ƙayyade yankin yankin da aka ɗaure da lanƙwasa y = x² + 1 da y = 2x + 2 a cikin tazara x = 0 zuwa x = 1.
Tattaunawa
Da farko, muna buƙatar nemo wuraren haɗuwa don tantance iyakokin haɗin. Maganin \( x^2 + 1 = 2x + 2 \):
\[ x^2 + 1 = 2x + 2 \]
\[ x^2 – 2x – 1 = 0 \]
Amfani da dabarar quadratic:
\[ x = \frac{2 \pm \sqrt{4 + 4}}{2} \]
\[ x = \frac{2 \pm \sqrt{8}}{2} \]
\[ x = \frac{2 \pm 2\sqrt{2}}{2} \]
\[ x = 1 \pm \sqrt{2} \]
Duk da haka, ga iyakoki na sama da na ƙasa tsakanin 0 da 1, ba ma buƙatar amfani da maganin quadratic ba, kawai iyaka ta yau da kullun daga 0 zuwa 1. Na gaba, ƙididdige yankin lanƙwasa na sama y ban da lanƙwasa na ƙasa bisa ga waɗannan iyakoki:
\[ L = \int_{0}^{1} [(2x + 2) – (x^2 + 1)] \, dx \]
Sauƙaƙa aiki:
\[ L = \int_{0}^{1} (2x + 2 – x^2 – 1) \, dx \]
\[ L = \int_{0}^{1} (-x^2 + 2x + 1) \, dx \]
Na gaba, mun sami maganin hana haihuwa:
Maganin hana \( (-x^2) \) shine \( -\frac{x^3}{3} \),
Maganin hana \( (2x) \) shine \( x^2 \),
Maganin hana \( (1) \) shine \( x \).
Don haka,
\[ L = \left. \left(-\frac{x^3}{3} + x^2 + x \right) \right|_0^1 \]
Kimantawa ta gaba:
\[ L = \left[ -\frac{1^3}{3} + 1^2 + 1 \right] – \left[ -\frac{0^3}{3} + 0^2 + 0 \right] \]
\[ L = \left[ -\frac{1}{3} + 1 + 1 \right] – \left[ 0 \right] \]
\[ L = -\frac{1}{3} + 2 \]
\[ L = \frac{6}{3} – \frac{1}{3} \]
\[ L = \frac{5}{3} \]
Don haka, yankin yankin da aka iyakance da lanƙwasa y = x² + 1 da y = 2x + 2 akan tazara [0, 1] shine:
\[ \frac{5}{3} \, \text{naúrar yanki} \]
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Daga misalan da ke sama, za mu iya ganin yadda za a iya amfani da haɗin gwiwa don ƙididdige yankin da ke ƙarƙashin lanƙwasa ko tsakanin lanƙwasa biyu. Tare da fahimtar ainihin ra'ayoyin haɗin gwiwa da dabarun hana rarrabuwa, ƙididdige waɗannan yankuna ya zama mai tsari da inganci. Da fatan, wannan labarin ya ƙara fahimtarmu game da amfani da haɗin gwiwa a cikin ainihin duniya, musamman a fannin auna yankin saman jirgin sama.
