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bmelyk
Express (limit given below) as a definite integral over [0,1] by first recognizing the indicated sum as Riemann Sum associated with a regular partition of [0,1] therefore over the interval [0,1]
\[\lim_{n \rightarrow 0} \frac{ 1^{3}+2^{3}+3^{3}+...+n^{3} }{ n^{4} }\]
\[\lim_{n \rightarrow \infty}***\]
\[\sum_{i=1}^n\frac{i^3}{n^4}=\sum_{i=1}^n\left(\frac{i}{n}\right)^3\frac{1}{n}\] \[\lim_{n\rightarrow \infty} \sum_{i=1}^n\left(\frac{i}{n}\right)^3\frac{1-0}{n};b=1,a=0\\=\int_0^1 x^3\;\mathrm dx\]