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