Question

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]

Answer 1

\[\lim_{n \rightarrow 0} \frac{ 1^{3}+2^{3}+3^{3}+...+n^{3} }{ n^{4} }\]

Answer 2

\[\lim_{n \rightarrow \infty}***\]

Answer 3

\[\sum_{i=1}^n\frac{i^3}{n^4}=\sum_{i=1}^n\left(\frac{i}{n}\right)^3\frac{1}{n}\]
\[\lim_{n\rightarrow \infty} \sum_{i=1}^n\left(\frac{i}{n}\right)^3\frac{1-0}{n};b=1,a=0\\=\int_0^1 x^3\;\mathrm dx\]