Answer
$Trapezoidal Rule$ = $4.2500$
$Simpson’s Rule$ = $4.0000$
$Exact$ = $4.0000$
Work Step by Step
$Trapezoidal Rule$
$\int _{0}^{2}$ $x^{3}$ $dx$ = $\frac{2}{8}$ $[f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+f(x_{4})]$
= $\frac{1}{4}$$[0+2(\frac{1}{8})+2(1)+2(\frac{27}{8})+8]$
= $\frac{1}{4}$$[0+\frac{1}{4}+2+\frac{27}{4}+8]$
= $\frac{17}{4}$
= $4.2500$
$Simpson’s Rule$
$\int _{0}^{2}$ $x^{3}$ $dx$ = $\frac{2}{12}$ $[f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+f(x_{4})]$
= $\frac{1}{6}$$[0+4(\frac{1}{8})+2(1)+4(\frac{27}{8})+8]$
= $\frac{1}{6}$$[0+\frac{1}{2}+2+\frac{27}{2}+8]$
= $\frac{24}{6}$
= $4.0000$
$Exact$
$\int _{0}^{2}$ $x^{3}$ $dx$ = $\frac{(2)^{4}}{4}$
= $4.0000$
