Evaluate `int0 1e^(2-3x)dx`as a limit of a sum.

Explanation

`" Let " f(x) =e^(2-3x)`
`" then " int_(0)^(1) e^(2-3x) dx=int_(0)^(1) f(x) dx`
we know that
`int_(a)^(b) f(x) dx =underset(h to 0)("lim") h[f(a)+f(a+h)`
`+f(a+2h)+.......+f(a+(n-1))h]`
`" where " nh =b-a`
`" here " a=0,b=1 " and " nh=1`
`" and " f(x)= e^(2-3x)`
`:. int_(0)^(1) e^(2-3x) dx=underset(h to0)("lim") h[f(0) +f(0+h)+`
`f(0+2h)+.......+f{0+(n-1)h}]`
`=underset(h to 0)("lim") h{e^(2)+e^(2-3h)+e^(2-6h) +......+ e^(2-3(h-1)h)}`
`=underset(h to0)("liom") he^(2) {1+e^(-3h)+e^(-6h)+....e^(-3(n-1)h)}`
`=underset(h to0)("lim") (he^(2) [1-(e^(-3h))^(n)])/(1-e^(-3h))=underset(h to 0)("lim") (e^(2)[1-{e^(-3(1))}])/((1-e^(-3h))/(-3h)(-3))`
`=(e^(2)(1-e^(-3)))/(underset(h to 0)(" 3 lim").(e^(-3h)-1)/(-3h))=(1)/(3)(e^(2)-e^(-1))`