Question

Using Riemann sums, find integral representations of the following:?

(1) `lim_(nrarroo)sum_(i=1)^n 2e^sqrt(i/n)1/n`
(2) `lim_(nrarroo)sum_(i=1)^n (i/n)^27 * 1/n`

Answer 1

Please see below for (1).

Explanation:

Using `x_i =`right endpoints, we have

`int_a^b f(x) dx = lim_(nrarroo)sum_(i=1)^n [f(a+i(b-a)/n)(b-a)/n]`

So `lim_(nrarroo)sum_(i=1)^n 2e^sqrt(i/n)1/n`

has `f(a+i(b-a)/n) = 2e^sqrt(i/n)` and `(b-a)/n - 1/n`

So `(b-a)/n = 1/n` and `a+i(b-a)/n = i/n = i(1/n)`, so `a = 0`.

Observe that when `i = n`, we have `a+i(b-a)/n = b`.
In this case we get `b = n/n = 1`

We have

`int_a^b f(x) dx = lim_(nrarroo)sum_(i=1)^n [f(a+i(b-a)/n)(b-a)/n]`

`lim_(nrarroo)sum_(i=1)^n 2e^sqrt(0+i/n) 1/n`

So `f(x) = 2e^sqrtx`

and we get

`lim_(nrarroo)sum_(i=1)^n 2e^sqrt(i/n)1/n = int_0^1 2e^sqrtx dx`

Answer 2

Please see below for (2).

Explanation:

Using `x_i =`right endpoints, we have

`int_a^b f(x) dx = lim_(nrarroo)sum_(i=1)^n [f(a+i(b-a)/n)(b-a)/n]`

`= lim_(nrarroo)sum_(i=1)^n i^27/n^27 * 1/n`

`= lim_(nrarroo)sum_(i=1)^n (i/n)^27 * 1/n`

`f(x) = x^27`, `a = 0` and `b = 1`, so

`int_0^1 x^27 dx`