InterviewSolution
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InterviewSolution
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योग परिभाषा द्वारा `int_(a)^(b)x^(2)dx` का मान ज्ञात कीजिए । |
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Answer» `int_(a)^(b)x^(2)dx=underset(hrarr0)(lim)h[a^(2)+(a+h)^(2)+(a+2h)^(2)+…+(a+(n-1)h)^(2)]` `" "` जहाँ `nh=b-a` `=underset(hrarr0)(lim)[(nh)a^(2)+a(nh)(nh-h)+(1)/(6)(nh)(nh-h)(2nh-h)]` `=underset(hrarr0)(lim)[(b-a)a^(2)+a(b-a)(b-a-h)+(1)/(3)(b-a-h)(b-a-(h)/(2))(b-a)]` `=(b-a)a^(2)+(b-a)^(2)a+(1)/(3)(b-a)^(3)` `=((b-a)/(3))[3a^(2)+3a(b-a)+(b-a)^(2)]` `=(b-a)/(3)[3a^(2)+3ab-3a^(2)+b^(2)-2ab+a^(2)]` `=(b-a)/(3)(a^(2)+ab+b^(2))` `=(1)/(3)(b^(3)-a^(3))` |
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