Answer
$Trapezoidal Rule$ = $2.7500$
$Simpson’s Rule$ = $2.6667$
$Exact$ = $2.6667$
Work Step by Step
$Trapezoidal Rule$
$\int _{0}^{2}x^{2}$ $dx$ = $\frac{2}{8}$ $[f(x_{0})+2f(x_{1})+2f(x_{2})+2f(x_{3})+f(x_{4})]$
$\frac{2}{8}$$[0+2(\frac{1}{4})+2(1)+2(\frac{9}{4})+4]$
= $\frac{1}{4}$$[0+\frac{1}{2}+2+\frac{9}{2}+4]$
= $\frac{11}{4}$
= $2.7500$
$Simpson’s Rule$
$\int _{0}^{2}x^{2}$ $dx$ = $\frac{2}{12}$ $[f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+f(x_{4})]$
= $\frac{2}{12}$$[0+4(\frac{1}{4})+2(1)+4(\frac{9}{4})+4]$
= $\frac{8}{3}$
= $2.6667$
$Exact$
$\int _{0}^{2}x^{2}$ $dx$ = $\frac{(2)^{3}}{3}$ - $0$
= $\frac{8}{3}$
= $2.6667$
