योगफल कि सीमा के रूप में ` int _(0)^(1) e^(x) dx ` का मान ज्ञात कीजिए ।

Explanation

परिभाषा से , ` int _(a)^(b) f(x) = lim_(h to 0) sum _(r =1)^(n) (a+rh)" "` …(1)
जहाँ, `nh = b-a` तथा `nto oo` यहाँ ,`f(x) = e^(x) ,a=0 ` तथा b=1
` :. f(a+rh) = f(0+rh) = f(rh) = e^(rh)`
तथा ` nh = b - a = 1 - 0 = 1 `
अब ( 1) से , ` int _(0)^(1) f(x) dx = lim _(h to 0) h sum_(r=1)^(n) f(a+rh) = lim_(nto0) h sum _(r=1)^(n) e^(rh)`
` = lim _(h to 0 ) h (e^(h) +e^(2h)+e^(3h)+...+ e^(nh))`
` = lim _(h to 0 ) h (e^(h)(1-e^(nh)))/(1-e^(h)) h (e^(h)(1-e^(1)))/(1-e^(h))" " [ :' nh = 1 ] `
` = lim _(hto 0) he^(h) ((e-1))/(e^(h)-1)= lim_(h to 0) e^(h) (e-1) 1/((e^(h)-1)/h)`
= ` e - 1 " " [ :' lim _(h to 0 ) (e^(h)-1)/h=1]`