Express the limit as a definite integral.

Question

please show the whole process step by step so i can understand the problem.

Answer 1

Divide the interval [0, 1] into n equal subintervals, each of length Δx = (1-0)/n = 1/n

The right endpoints of the subintervals are 1/n, 2/n, 3/n, ...,n/n=1

Let x_i = right endpoint of interval i = i / n

Let f(x) = 3 / (1 + x⁶).

For each of the n subintervals, multiply f(x_i) by Δx to get:

f(x₁)Δx + f(x₂)Δx + f(x₃)Δx + ... f(x_n)Δx

= Δx [ f(x₁) + f(x₂) + ...+f(x_n) ] = (1/n)∑_{(I = 1 to n)} f(i /n) = (1/n)∑_{(I = 1 to n)} [ 3 / (1 + (i/n)⁶)]

The value of the limit of the sum above as n→∞ is the value of:

∫_{(from 0 to 1)} [3 / (1 + x⁶)] dx