InterviewSolution
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InterviewSolution
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` int _(1)^(3)x^(3)dx` का मान ज्ञात कीजिए । |
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Answer» परिभाषा से , ` int _(a)^(b) f(x) dx = lim _(h to 0) sum_(r=1)^(n) h f (a+rh)" "` …(1) जहाँ, `nh = b-a` तथा `n to oo` यहाँ a = 1, b=3 ` :. nh = b- a = 3 -1 = 2` ` f(x) = x^(3) :. f(a+rh) = f(1+rh) = (1+rh)^(3)` ` = 1^(3) +3rh + 3r^(2) h^(2) + r^(3) h^(3)` अब (1 ) से , ` int _(1)^(3) x^(3) dx = lim _(h to 0 ) sum_(r=1)^(n) h f(a+rh)` ` = lim _(h to 0 ) sum_(r=1)^(n) h (1^(3) + 3rh + 3h^(2) r^(2) +h^(3)r^(3))` ` = lim _(h to 0 ) ( h sum _(r=1)^(n) 1+ 3h^(2) sum_(r=1)^(n)r + h^(4) sum_(r=1)^(n) r^(3))` ` = lim _(h to 0) [ nh + 3h^(2) (n(n+1))/2 3 h^(2) + (n(n+h)(2n+1))/1 + h^(2) (n^(2)(n+1)^(2))/4 ] ` ` = lim _( h to 0 ) [ nh + 3/2 nh (nh+h) + (nh(nh+h)(2nh+h))/2 + ((nh)^(2)(nh+h)^(2))/4]` ` lim _( h to 0 ) [ 2 + 3/2 xx 2 (2+h) + (2(2+h)(4+h))/h + ((2)^(2).(2+h)^(2))/4] = 20 ` |
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