Question

Evaluate the following integrals as the limit of the sum:

`int_0^1 x^2  "d"x`

Answer 1

`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`