Chapter 8.6, Problem 44E

Approximating a Function The table lists several measurements gathered in an experiment to approximate an unknown continuous function $y=f\left(x\right)$.

x	0.00	0.25	0.50	0.75	1.00
Y	4.32	4.36	4.58	5.79	6.14

x	1.25	1.50	1.75	2.00
y	7.25	7.64	8.08	8.14

Approximate the integral

${\displaystyle {\int }_0^{2}f\left(x\right)dx}$

Using the Trapezoidal Rule and Simplson’s Rule.

Use a graphing utility to find a model of the form $y=a{x}^3+b{x}^2+cx+d$ for the data. Integrate the resulting polynomial over $\left[0,2\right]$ and compare the result with the integral from part (a).