Find the exact value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).
Use Simpson's rule with two strips to show that an approximate value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\) can be expressed in the form \(m + n \sqrt [ 3 ] { 36 }\), where the values of the constants \(m\) and \(n\) are to be stated.
Use the results from parts (i) and (ii) to find an approximate value of \(\sqrt [ 3 ] { 36 }\), giving your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
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