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

Question

(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#

Christopher Allers · Accepted Answer

Please see below for (1).

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#

Paisley Johnson · Answer

Please see below for (2).

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#

Emilia Aguilar · Answer

undefined Sure, I'd be happy to help. Could you please provide the function or the expression for which you'd like to find integral representations using Riemann sums?

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#

(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#