योगफल कि सीमा के रूप में ` int _(0)^(2) 2^(x) dx ` का मान ज्ञात कीजिए ।

Explanation

` int _(a)^(b) f(x) dx = lim _(h to 0) sum _(r=1)^(n) h f (a+rh)" "` … (1)
जहाँ, `nh=b-a ` तथा `nto oo` यहाँ `f(x) = 2^(x) , a = 0, b=2`
तथा ` nh = b-a = 2- 0 = 2 ` साथ ही ` f(a+rh) = f(0+rh) = f(rh)= 2^(rh)`
अब ( 1) से ,
` int _(0)^(2) 2^(x) dx = lim _(h to 0) sum_(r=1)^(n) h2^(rh)`
` lim _(h to 0) h(2^(h) +2^(2h)+...+ 2^(nh))`
` = lim _( h to 0) h. (2^(h)[(2^(h))^(n)-1])/(2^(h)-1) " " [ :' " G.P के n पदों का योगफल "= (a(r^(n)-1))/(r-1)]`
` = lim _( h to 0) h. 2^(h) ((2^(nh)-1)/(2^(h)-1)) = lim _(h to 0 ) h.2^(h) ((2^(2)-1)/(2^(h)-1))`
` = lim _(h to 0 ) .2^(h) (3/((2^(h)-1)/h)) = 3/(log_(e)2)" " [ :' lim_( h to 0) (a^(x)-1)/x = log_(e)a]`