Formula for polynomial function / Finding Area under the Curve

[#1]

So, I would like to find the Area under some curves. Actually quite many of them, so using the Trapezoid method manually won't work, and can't imagine it would be valid enough. There is some kind of function for this for ROC-curves, but is it usable for finding the AUC in curves, measureing blood substances over time (five times)?

However, if I get a formula for a fitted Curve, I can use it to find AUC (Integral) in other software. For example, a Sixth degree polynomial. Here, I get the the function but I can't make sense of it (pic 1). 

Polynomial functions are:
But I can't make sense of the Formula JMP provides me. Using only Time=3? 

By using a flexible smoothing spline fit, I can get some coeffecients, but those won't help me (pic 2)? 



Would be grateful for tips getting the equation integral-ready and/or finding the AUC easily, for many subjects/curves.

[#2]

Good point, Craige.

The centering also minimizes the correlations between the estimates. Otherwise, the correlations increase the standard errors of the estimates. The larger standard errors produce wider confidence interval estimates and smaller t-ratios for the same estimate. The smaller t-ratio will have a higher p-value. So correlations compromise the power of the decision about the estimate being significantly different from zero.