Chapter 4 - Integration - 4.4 The Definition Of Area As A Limit; Sigma Notation - Exercises Set 4.4 - Page 298: 16

\[ -420 \]

Remember that: \[ \sum_{k=1}^{n} k^{3}=\left[\frac{(1+n)n}{2}\right]^{2} \] and \[ \sum_{k=1}^{n} k=\frac{(1+n)n}{2} \] We find: \[ \sum_{k=1}^{6} k-k^{3}=\sum_{k=1}^{6} k-\sum_{k=1}^{6} k^{3}=\left[\frac{(1+6)6}{2}\right]-\left[\frac{(1+6)6}{2}\right]^{2}=-420 \]