limit of summation notation of (i^4/n^5+i/n^2) when n goes to infinity on interval[0,1]

Question

evaluate limit by recognizing it as a Riemann sum for a function f(x) on the interval [0,1],and then using integration

limit(n to inf) sigma(i^4/n^5+i/n^2) ,,(i from 1 to inf)

my teacher said that we should use f(x)=x^4+x ,then we should take integral of this function on the interval [0,1]

but I want to know how we found f(x)??

Answer 1

Suppose you knew the function was f(x) = x⁴ +x and had to calculate the area under the curve from [01] using left-hand Riemann sum, dividing the interval into n equal subintervals and then letting n->∞.

The length of each sub-interval would be (1-0)/n or 1/n.

What would a formula look like for the area of each rectangle.

x_{1 = 0 + 1/n}

x₂ = 0 + 2/n

...

x_{i= 0 + i/n.}

Area of ith rectangle:

R₁ = 1/n [f(1/n)] = 1/n[1/n⁴ + 1/n]

R_{2 = 1/n[f(2/n)] = 1/n[(2/n)}⁴ _{+ (2/n)}]

R_{i = 1/n[f(i/n)] = 1/n[(i/n)}⁴_+(i/n)]

_{Distribute that 1/n for R sub i,}

_{To answer the original question picture working backwards, factoring out the 1/n, etc. A more interesting challenge would be to actually evaluate the Riemann sum by taking the limit using formulas for the sum from i = i to n of i and i^4.}