Question

Evaluate the following integrals as the limit of the sum:

`int_1^3 x  "d"x`

Answer 1

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