Question

Evaluate the following integrals as the limit of the sum:

`int_1^3 (2x + 3)  "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 = 1

 b = 3

h = `("b" - "a")/"n"`

= `(3 - 1)/"n"`

= `2/"n"`

fx = (2x + 3)

f(a + rh) = `"f"(1 + 2/"n" ("r"))`

`int_1^3 "f"(x)  "d"x = lim_("n" ->  oo) sum_("r" = 1)^"n" 2/"n" [2(1 + (2"")/"n") + 3]`

= `lim_("n" ->  oo) sum_("r" = 1)^"n" 2/"n" (2 + (4"r")/"n" + 3)`

= `lim_("n" ->  oo) sum_("r" = 1)^"n" 2/"n" (5 + 4/"n" "r")`

= `lim_("n" ->  oo) sum_("r" = 1)^"n" (10/"n" + 8/"n"^2 ("r"))`

= `lim_("n" ->  oo) [10/"n" sum(1) + 8/"n"^2 sum("r")]`

= `lim_("n" ->  oo) [10/"n" ("n") + 8/"n"^2 (("n"("n" + 1))/"n")]`

= `lim_("n" ->  oo) [10 + 4(1 + 1/"n")]`

= 10 + 4(1 + 0)

= 10 + 4

= 14