Problem 2(40pts): We can calculate the volume of a system as V=abP(x)dx.P(x) function values against x values were obtained experimentally as follows: a) Calculate the system volume (V) using Simpson's 1/3 rule (2 segments application). (20 pts) (Correct solution set up is 10pts and correct result is 10pts ) b) If f(x)=(V1.85792e311x)2.05, find the root of the function using the Newton-Raphson Method. (6 significant digits solution, s=103 and x0=0.5 ) Note: V is calculated in part a. (20 pts) (Correct solution set up is 10pts and correct results are 10pts) Appendix: \begin{tabular}{l} Table 2.9. Algorithm of the Newton-Raphson method \\ \hline Step 1: Estimate an initial value of x0 close to the root. \\ Step 2; Calculate f(x) and f(xi). \\ Step 3: xi+1=xif(xi)f(xi) Calculate a new value for the root from this equation. \\ Step 4: if a=xi+1xi+1xjx100,, then stop, otherwise return the step 2 by assigning xi=xi+1 \\ \hline \end{tabular} Linear Regression: Normal equations that can be solved simultaneously for a1 and a0 : a1=nxi2(xi)2nxiyixiyiao=nyia1nxi=ya1x Polynomial Regression: (n)a0+(xi)a1+(xi2)a2(xi)a0+(xi2)a1+(xi3)a2(xi2)a0+(xi3)a1+(xi4)a2=yi=xiyi=xi2yi Simpson 1/3 Rule Classical Fourth-Order Runge-Kutta Method - The most popular RX methods are fourth-order, and the most - Subtitute into the Intsyral for ff(x) and commonly used form is. integrale: f=3h{f(xn)+4f(x1)+f(x2)}withh=2ba 1,f(x,y)i2=f(x+21knx1+21k1,k)