Evaluate: `int_a^b e^x dx` using limit of sum

Explanation

Here `f(x)=e^(x)`.
`:.int_(a)^(b)e^(x)=lim_(nto oo)h. sum_(r=0)^(n-1)e^(a+rh)`
`=lim_(nto oo) [e^(a)+e^(a+h)e^(a+2b)+……+e^(a+((n-1)h))]`
where `b-a=nh`
`=lim_(nto oo) he^(a)[1+e^(h)+e^(2h)+……….+e^((n-1)h)]`
`=lim_(n to oo) he^(a)[{(e^(h))^(n)-1}//(e^(h)-1)]` (sum of G.P.) ltbgt `=lim_(h to oo) e^(a)(e^(nh)-1).[h//(e^(h)-1)]`
`[ :' "as" n to oo,h to 0, "and" nh=b-a]`
`=e^(a)(e^(b-a)-1)1=e^(b)-a^(a)`