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प्रश्न
Evaluate the following integrals as the limit of the sum:
`int_0^1 x^2 "d"x`
योग
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उत्तर
`int_"a"^"b" "f"(x) "d"x = lim_(("n" -> oo)/("h" -> 0)) sum_("r" = 1)^"n" "hf" ("a" + "rh")`
Here a = 0
b = 1
H = `("b" - "a")/"n"`
= `(1 - 0)/"n"`
= `1/"n"`
f(x) = x2
f(a + rh) = `"f"(0 + "r"(1/"n"))`
= `"f"("r"/"n")`
`int_0^1 "f"(x) "d"x = lim_("n" -> oo) sum_("r" = 1)^"n" ("r"/"n")^2`
- `lim_("n" -> oo) sum_("r" = 1)^"n" "r"^2/"n"^3`
= `lim_("n" -> oo) [1/"n"^3 sum "r"^2]`
= `lim_("n" -> oo) [1/"n"^3 (("n"("n" + 1)(2"n" + 1))/6)]`
= `lim_("n" -> oo) [1/6 (("n" + 1))/"n" xx ((2"n" + 1))/3]`
= `lim_("n" -> o)[1/6 xx (1 + 1/"n") xx (2 + 1/"n")]`
= `[1/6 xx (1) xx (2)]`
= `2/6`
= `1/3`
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