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Question
Evaluate the following as limit of sum:
`int_0^2 "e"^x "d"x`
Solution
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
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