Question

निम्नलिखित का योग की सीमा के रूप में मान निकालिए-

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

Answer 1

हम जानते हैं कि `int_"a"^"b" "f"(x)"d"x = lim_("n"-> oo) "h" sum_("r" = 0)^("n" - 1) "f"("a" + "rh")`

I = `int _0^2 "e"^x "d"x` के लिए

हमारे पास a = 0 और b = 2 है।

∴ "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) "h" (("e"^2 - 1)/("e"^"h" - 1))`

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

= `"e"^2 - 1`