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Question
Evaluate the following integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
Solution
`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
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