Su'aalo iyo Doodo Tusaalo ah oo ku saabsan Adeegsiga Isku-dhafka ee Xisaabinta Aagga Diyaarad Fidsan
Baridda xisaabta, isku-dhafka waxaa badanaa lagu arkaa xisaabinta. Mid ka mid ah codsiyada ugu caansan ee isku-dhafka waa xisaabinta aagga hoos yimaada qalooca ama diyaaradda. Maqaalkani wuxuu ka hadli doonaa dhowr dhibaato oo tusaale ah wuxuuna ka hadli doonaa isticmaalka isku-dhafka si loo xisaabiyo aagga diyaaradda.
Hordhac ku saabsan Aragtida
Kahor inta aan u gudubno dhibaatada tusaalaha ah, aan dib u eegno fikradda aasaasiga ah ee xisaabinta aagga hoos yimaada qalooca iyadoo la adeegsanayo isku-dhafan. Haddii aan leenahay shaqo f(x) ah oo joogto ah oo ku taal muddada [a, b], markaa aagga ka hooseeya qalooca y = f(x) laga bilaabo x = a ilaa x = b waa:
\[ L = \int_{a}^{b} f(x) \, dx \]
Juqraafi ahaan, tani waxay ka dhigan tahay inaan soo koobeyno bedka leydi aad u khafiif ah laga bilaabo x = a ilaa x = b.
Su'aal Tusaale 1aad
Soal
Xisaabi bedka ka hooseeya qalooca y = x² ee ku jira farqiga [1, 3].
Dood
Si loo xisaabiyo bedka, waxaan isticmaalnaa isku-darka:
\[ L = \int_{1}^{3} x^2 \, dx \]
Waxaan ku bilaabaynaa inaan helno lidka ku ah \( x^2 \). Lidka ku ah \( x^2 \) waa \( \frac{x^3}{3} \). Kadibna isku-dhafka ayaa noqonaya:
\[ L = \left[ \frac{x^3}{3} \right]_{1}^{3} \]
Xasuuso inaan qiimeyno lidka-soo-saarka xadka isku-dhafka ah:
\[ L = \left( \frac{3^3}{3} \right) – \left( \frac{1^3}{3} \right) \]
\[ L = \left( \frac{27}{3} \right) – \left( \frac{1}{3} \right) \]
\[ L = 9 – \frac{1}{3} \]
\[ L = \frac{27}{3} – \frac{1}{3} \]
\[ L = \frac{26}{3} \]
Markaa, aagga ka hooseeya qalooca y = x² laga bilaabo x = 1 ilaa x = 3 waa:
\[ \frac{26}{3} \, \text{cutubka aagga} \]
Su'aal Tusaale 2aad
Soal
Go'aami bedka gobolka ee ay ku xiran tahay qalooca y = x³ iyo xariiqyada x = 1 iyo x = 2.
Dood
Si loo xisaabiyo bedka, waxaan isticmaalnaa isku-darka:
\[ L = \int_{1}^{2} x^3 \, dx \]
Sida caadiga ah, waxaan ku bilaabaynaa helitaanka lidka ku ah \( x^3 \). lidka ku ah \( x^3 \) waa \( \frac{x^4}{4} \). Isku-dhafka wuxuu noqonayaa:
\[ L = \left[ \frac{x^4}{4} \right]_{1}^{2} \]
Qiimee xadka isku-dhafka ah:
\[ L = \left( \frac{2^4}{4} \right) – \left( \frac{1^4}{4} \right) \]
\[ L = \left( \frac{16}{4} \right) – \left( \frac{1}{4} \right) \]
\[ L = 4 – \frac{1}{4} \]
\[ L = \frac{16}{4} – \frac{1}{4} \]
\[ L = \frac{15}{4} \]
Markaa, aagga ka hooseeya qalooca y = x³ laga bilaabo x = 1 ilaa x = 2 waa:
\[ \frac{15}{4} \, \text{cutubka aagga} \]
Su'aal Tusaale 3aad
Soal
Go'aami bedka gobolka ee ay ku xiran yihiin qaloocyada y = x² + 1 iyo y = 2x + 2 muddada x = 0 ilaa x = 1.
Dood
Marka hore, waxaan u baahanahay inaan helno meelaha isgoysyada si aan u go'aamino xadka isku-dhafka. Xalka \( x^2 + 1 = 2x + 2 \):
\[ x^2 + 1 = 2x + 2 \]
\[ x^2 – 2x – 1 = 0 \]
Iyadoo la isticmaalayo qaacidada labajibbaaran:
\[ x = \frac{2 \pm \sqrt{4 + 4}}{2} \]
\[ x = \frac{2 \pm \sqrt{8}}{2} \]
\[ x = \frac{2 \pm 2\sqrt{2}}{2} \]
\[ x = 1 \pm \sqrt{2} \]
Si kastaba ha ahaatee, xadka sare iyo kan hoose ee u dhexeeya 0 iyo 1, uma baahnin inaan isticmaalno xalka labajibbaaran, kaliya xadka isku-dhafka ah ee caadiga ah laga bilaabo 0 ilaa 1. Marka xigta, xisaabi bedka qalooca y ee sare laga jaray qalooca y ee hoose iyadoo loo eegayo xadkan:
\[ L = \int_{0}^{1} [(2x + 2) – (x^2 + 1)] \, dx \]
Fududeynta shaqada:
\[ L = \int_{0}^{1} (2x + 2 – x^2 – 1) \, dx \]
\[ L = \int_{0}^{1} (-x^2 + 2x + 1) \, dx \]
Marka xigta, waxaan helnaa waxyaabaha lidka ku ah:
Ka-hortagga \( (-x^2) \) waa \( -\frac{x^3}{3} \),
Ka-hortagga \( (2x) \) waa \( x^2 \),
Ka-hortagga \( (1) \) waa \( x \).
Markaa,
\[ L = \left. \left(-\frac{x^3}{3} + x^2 + x \right) \right|_0^1 \]
Qiimaynta xigta:
\[ L = \left[ -\frac{1^3}{3} + 1^2 + 1 \right] – \left[ -\frac{0^3}{3} + 0^2 + 0 \right] \]
\[ L = \left[ -\frac{1}{3} + 1 + 1 \right] – \left[ 0 \right] \]
\[ L = -\frac{1}{3} + 2 \]
\[ L = \frac{6}{3} – \frac{1}{3} \]
\[ L = \frac{5}{3} \]
Markaa, bedka gobolka ee ay ku xiran yihiin qaloocyada y = x² + 1 iyo y = 2x + 2 marka loo eego farqiga [0, 1] waa:
\[ \frac{5}{3} \, \text{cutubka aagga} \]
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Tusaalooyinka kor ku xusan, waxaan ka arki karnaa sida isku-dhafka loo isticmaali karo in lagu xisaabiyo aagga hoos yimaada qalooca ama inta u dhaxaysa laba qalooc. Iyada oo si habboon loo fahmo fikradaha aasaasiga ah ee isku-dhafka iyo farsamooyinka ka-hortagga, xisaabinta meelahan waxay noqotaa mid aad u nidaamsan oo hufan. Waxaan rajeyneynaa, maqaalkani wuxuu kordhiyay fahamkeenna ku saabsan isticmaalka isku-dhafka adduunka dhabta ah, gaar ahaan dhinaca cabbirka bedka dusha sare ee diyaaradda.
