Which of the following correctly expresses the limit $\displaystyle \lim_{n \rightarrow \infty} \sum_{i=1}^n \frac{i^4}{n^5},$ as a definite integral?

A. $\displaystyle \int_{0}^{\,1} {x^3}\,dx$
B. $\displaystyle \int_{1}^{\,2} {x^3}\,dx$
C. $\displaystyle \int_{0}^{\,1} {x^4}\,dx$
D. $\displaystyle \int_{0}^{\,1} {x^2}\,dx$
E. $\displaystyle \int_{1}^{\,2} {x^2}\,dx$
F. $\displaystyle \int_{1}^{\,2} {x^4}\,dx$