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प्रश्न
Evaluate the following integrals as the limit of the sum:
`int_1^3 x "d"x`
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उत्तर
We have `int_"a"^"b" "f"(x) "d"x = lim_(("n" -> oo),("h" -> 0)) sum_("r" = 1)^"n" "hf" ("a" + "rh")`
Here a = 1
b = 3
h = `("b" - "a")/"n"`
= `(3 - 1)/"n"`
= `2/"n"`
f(x) = x
f(a + rh) `"f"(1+ (2"r")/"n")`
`int_1^3 x "d"x = lim_("n" -> oo) sum_("r"= 1)^"n" 2/"n"(1 + (2"r")/"n")`
= `lim_("n" -> oo) sum_("r" = 1)^"n" (2/"n" +(4"r")/"n"^2)`
= `lim_("n" -> oo) [2/"n" sum1 + 4/"n"^2 sum"r]`
= `lim_("n" -> oo) [2/"n" ("n") + 4/"n"^2 (("n"("n" + 1))/2)]`
= `lim_("n" -> oo) [2 + 2(1 + 1/"n")]`
= 2 + 2(1 + 0)
= 2 + 2
= 4
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