Tusaalooyinka Su'aalaha iyo Doodaha Isku-dhafka La Hubo
Isku-dhafka qeexan waa fikrad muhiim ah oo ku jirta kalkulus, oo inta badan loo isticmaalo in lagu helo aagga hoos yimaada qalooca, lagu xisaabiyo mugga walxaha adag, iyo codsiyo kale oo badan oo ku saabsan injineernimada iyo fiisigiska. Ka hadalka isku-dhafka sugan ma aha oo kaliya inuu bixiyo faham aasaasi ah oo ku saabsan fikraddan laakiin sidoo kale wuxuu xoojiyaa xirfadaheena falanqaynta xisaabta. Maqaalkani wuxuu higsanayaa inuu bixiyo tusaalooyin dhibaatooyin isku-dhafan oo qeexan oo ay weheliso doodo faahfaahsan.
Fikradda Aasaasiga ah ee Isku-dhafka La Hubo
Kahor inta aynaan guda gelin dhibaatooyinka tusaalaha ah, aan dib u eegno qaar ka mid ah fikradaha aasaasiga ah ee isku-dhafka qeexan. Isku-dhafka qeexan, oo lagu tilmaamay \(\int_a^bf(x) \, dx\), wuxuu matalaa aagga ka hooseeya qalooca shaqada \(f(x)\) laga bilaabo barta \(x = a\) ilaa barta \(x = b\).
Xisaab ahaan, isku-dhafka qeexan ee ka bilaabma \(a\) ilaa \(b\) ee shaqada \(f(x)\) waxaa loo qeexi karaa sidan:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
halkaas oo \(F(x)\) uu yahay lidka ku ah \(f(x)\).
Su'aalo iyo Doodo Tusaale ah
Aan eegno tusaalooyin ka mid ah dhibaatooyinka muhiimka ah iyo doodahooda.
Su'aal Tusaale 1aad
Su'aal:
Xisaabi isku-dhafka qeexan ee shaqada \(f(x) = 2x\) laga bilaabo \(x = 1\) ilaa \(x = 3\).
Dood:
Si aan u xallino isku-dhafkan, marka hore waxaan helnaa lidka ku ah \(f(x) = 2x\).
Ka-hortagga \(2x\) waa:
\[ F(x) = x^2 + C \]
Si kastaba ha ahaatee, isku-dhafanno qeexan uma baahnin isku-dhafan joogto ah \(C\).
Hadda, isticmaal xadka isku-dhafka si aad u xisaabiso:
\[ \int_1^3 2x \, dx = F(3) – F(1) \]
Xisaabi qiimaha \(F(x)\) ee xadkan:
\[ F(3) = 3^2 = 9 \]
\[ F(1) = 1^2 = 1 \]
Haddaba,
\[ \int_1^3 2x \, dx = 9 – 1 = 8 \]
Su'aal Tusaale 2aad
Su'aal:
Xisaabi isku-dhafka qeexan ee shaqada \(f(x) = x^2 + 1\) laga bilaabo \(x = 0\) ilaa \(x = 2\).
Dood:
Soo hel lidka ku ah \(f(x) = x^2 + 1\).
Ka-hortagga \(x^2\) waa:
\[ \frac{1}{3}x^3 \]
Ka-hortagga \(1\) waa \(x\).
Markaa, lidka ku ah \(f(x)\) waa:
\[ F(x) = \frac{1}{3}x^3 + x \]
Hadda, isticmaal xadka isku-dhafka si aad u xisaabiso:
\[ \int_0^2 (x^2 + 1) \, dx = F(2) – F(0) \]
Xisaabi qiimaha \(F(x)\) ee xadkan:
\[ F(2) = \frac{1}{3}(2)^3 + 2 = \frac{8}{3} + 2 = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \]
\[ F(0) = \frac{1}{3}(0)^3 + 0 = 0 \]
Haddaba,
\[ \int_0^2 (x^2 + 1) \, dx = \frac{14}{3} – 0 = \frac{14}{3} \]
Su'aal Tusaale 3aad
Su'aal:
Xisaabi isku-dhafka qeexan ee shaqada \(f(x) = e^x\) laga bilaabo \(x = 1\) ilaa \(x = 2\).
Dood:
Soo hel lidka ku ah \(f(x) = e^x\).
Ka-hortagga \(e^x\) waa \(e^x\).
Hadda, isticmaal xadka isku-dhafka si aad u xisaabiso:
\[ \int_1^2 e^x \, dx = F(2) – F(1) \]
Xisaabi qiimaha \(F(x)\) ee xadkan:
\[ F(2) = e^2 \]
\[ F(1) = e^1 = e \]
Haddaba,
\[ \int_1^2 e^x \, dx = e^2 – e \]
Su'aal Tusaale 4aad
Su'aal:
Xisaabi isku-dhafka qeexan ee shaqada \(f(x) = \sin(x)\) laga bilaabo \(x = 0\) ilaa \(x = \pi\).
Dood:
Soo hel lidka ku ah \(f(x) = \sin(x)\).
Ka-hortagga \(\sin(x)\) waa \(-\cos(x)\).
Hadda, isticmaal xadka isku-dhafka si aad u xisaabiso:
\[ \int_0^\pi \sin(x) \, dx = F(\pi) – F(0) \]
Xisaabi qiimaha \(F(x)\) ee xadkan:
\[ F (\pi) = -\cos (\pi) = -(-1) = 1 \]
\[ F(0) = -\cos(0) = -1 \]
Haddaba,
\[ \int_0^\pi \sin(x) \, dx = 1 – (-1) = 1 + 1 = 2 \]
Su'aal Tusaale 5aad
Su'aal:
Xisaabi isku-dhafka qeexan ee shaqada \(f(x) = \frac{1}{x}\) laga bilaabo \(x = 1\) ilaa \(x = e\).
Dood:
Soo hel lidka ku ah \(f(x) = \frac{1}{x}\).
Ka-hortagga \(\frac{1}{x}\) waa \(\ln|x|\).
Hadda, isticmaal xadka isku-dhafka si aad u xisaabiso:
\[ \int_1^e \frac{1}{x} \, dx = F(e) – F(1) \]
Xisaabi qiimaha \(F(x)\) ee xadkan:
\[ F(e) = \ln(e) = 1 \]
\[ F(1) = \ln(1) = 0 \]
Haddaba,
\[ \int_1^e \frac{1}{x} \, dx = 1 – 0 = 1 \]
Gabagabo
Tusaalooyinka kor ku xusan, waxaan ku tababarannay helitaanka isku-dhafan qeexan oo ka kooban shaqooyin kala duwan oo aasaasi ah. Tallaabo kasta, waxaa muhiim ah in marka hore la helo anti-derivative ka dibna la isticmaalo xadka isku-dhafka si loo helo qiimaha ugu dambeeya.
Isku-dhafka la hubo ayaa door muhiim ah ka ciyaara dhinacyo badan oo waxbarasho iyo adeegsiyo wax ku ool ah. Fahmidda fikraddan iyo ku celcelinta tusaalooyin kala duwan ayaa si weyn u xoojin doonta xirfadahaaga xisaabta.
