InterviewSolution
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InterviewSolution
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योगफल की सिमा के रूप में `int_(0)^(1) e^(2-3x)dx` का मान ज्ञात कीजिएः |
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Answer» यहाँ `0,b=1, f(x)=e^(2-3x),h(b-a)/(n)=(1)/(n)"या "nh=1` योगफल की सिमा की परिभाषा से, `underset(a)overset(b)int f(x)dx` `=underset(h to 0)limh[f(a)+f(a+h)+f(a+2h)+.....+f{a+(n-1)h}]` `=underset(h to 0)lim h[f(0)+f(0+h)+f(0+2h)+.....+f{a+(n-1)h}]` `=underset(h to 0)lim h[f(0)+f(h)+f(2h)+.....+f{a+(n-1)h}]` `=underset(h to 0)lim f[e^(2)+2^(2-3h)+e^(2-3(2h))+....+e^(2-3(n-1)h)]` `=underset(h to 0)lim he^2[1+e^(-3h)+e^(-3(2h))+....+e^(-3(n-1)h)]` `=underset(h to 0)lim he^2[1+e^(-3h)+(e^(-3h))^(2)+....+(e^(-3h))^(n-1)]` `=underset(h to 0)lim he^2[((e^(-3))^(n)-1)/(e^(-3h)-1)]=underset(h to 0)lim he^(2)[(e^(-3nh)-1)/(e^(-3h)-1)]` `=underset(h to 0)lim he^2[(e^(-3)-1)/(((e^(-3h)-1)/(h)))]` `=e^(2)xx(e^(-3)-1)xx(1)/(underset(h to 0)lim((e^(-3h)-1)/(-3)h)xx-3)` `=e^(2)(e^(-3)-1)xx(1)/(1xx(-3)) " "[therefore underset(x to 0)lim (e^(x)-1)/(x)=1]` `=-1/3(e^(-1)-e^(2))=1/3(e^(2)-e^(-1))` |
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