Question

Evaluate the following integrals as the limit of the sum:

`int_0^1 (x + 4)  "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 = 0

b = 1

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

= `(1 - 0)/"n"`

= `1/"n"`

f(x) = x + 4

f(a + rh) = `"f"(0 + "r"/"n")`

= `"f"("r"/"n")`

`int "f"(x)  "d"x = lim_("n" ->  oo) sum_("r" = 1)^"n"  1/"n" ("r"/"n" + 4)`

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

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

= `lim_("n" ->  oo) 1/"n"^2 [("n"("n" + 1))/2] + 4/"n" ("n")]`

= `lim_("n" ->  oo) [1/2[1 + 1/"n"] + 4]`

= `1/2[1 + 0] + 4`

= `1/2 + 4`

= `9/2`