InterviewSolution
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InterviewSolution
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योगफल की सिमा के रूप में `int_(1)^(3) x^(3)dx` का मान ज्ञात कीजिएः |
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Answer» यहाँ `a=1, b=3, f(x)=x^(3), nh=b-a=2` योगफल की सिमा की परिभाषा से, `underset(a)overset(b)int f(x)dx=underset(h to 0)limh[f(1)+f(1+h)+f(1+2h)+....+f(1+(n-1)h}]....(1)` अब `f(1)=(1)^(3)=1` `f(1+h)=(1+h)^(3)=1+h^(3)+3h+3h^(2)` `=1+3h+3h^(2)+h^(3)` `f(2+h)=(1+2h)^(3)=1+3(2h)+3(2h)^(2)+(2h)^(3)` `=1+6h+12h^2+8h^3` `f[1+(n-1)h]=1+3(n-1)h+3(n-1)^2h^2+(n-1)^(3)h^(3)` उपरोक्त मनो को समी में रखने पर `underset(1)overset(3)intx^(3)dx` `=underset(h to 0)lim h{1+(1+3h+3h^(2)+h^(3)) +(1+6h+12h^(2)+8h^(3)) +{1+3(n-1)h+3(n-1)^(2)h^(2)+(n-1)^(3)h^(3)}]` `=underset(h to 0)lim h[1+1+......+1("n बार ") +3h{1+2+......+(n-1)} +3h^(2){1^(2)+2^(2)+...+(n-1)^(2)} +h^(3){1^3+2^3+....+(n-1)^(3)}]` `=underset(h to 0)lim h[n+3hxx(n(n-1))/(2)+3h^(2)xx(n(n-1)(2n-1))/(6)+h^(3)xx {(n(n-1))/(2)}^(2)}` `=underset(h to 0)lim [nh+(3xxnh(nh-h))/(2)+(3xxnh(nh-h)(2nh-h))/(6)+{(nh(nh-h))/(2)}^2]` `=[2+(3xx2xx(2-0))/(2)+(3xx2xx(2-0)(2xx2-0))/(6)+{(2(2-0))/(2)}^(2)]` =2+6+8+4=20 |
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