Question

Evaluate the following as limit of sum:

`int_0^2 "e"^x "d"x`

Answer 1

We know that `int_"a"^"b" "f"(x)"d"x = lim_("n" -> oo) "h" sum_("r" = 0)^("n" - 1) "f"("a" + "rh")`

For I = `int_0^2 "e"^x "d"x`

We have a = 0 and b = 2

∴ h = `("b" - "a")/"n" = (2 - 0)/"n" = 2/"n"`

∴ I = `int_0^2 "e"^x "d"x`

= `lim_("h" -> 0) "h" [1 + "e"^"h" + "e"^(2"h") + ... + "e"^(("n" - 1)"h")]`

= `lim_("h" -> 0) "h" [(1 * ("e"^"h")^"n" - 1)/("e"^"h" - 1)]`

= `lim_("h" -> 0) "h" (("e"^("nh") - 1)/("e"^"h" - 1))`

= `lim_("h" -> 0) (("e"^2 - 1)/("e"^"h" - 1))`

= `"e"^2 lim_("h" -> 0) "h"/("e"^"h" - 1)`

= e² – 1