Use (a) the Midpoint Rule and (b) Simpson's Rule to approximate the integral $\int_{0}^{\,\pi} {x^2 \sin(x)}\, dx$ with $n = 8$. Give each answer correct to six decimal places.

$M_8 =$

$S_8 =$

(c) Evaluate the integral $\int_{0}^{\,\pi} {x^2 \sin(x)}\, dx$ and use it to find the absolute value of the error (to six decimal places) in each approximation from above.

$\int_{0}^{\,\pi} {x^2 \sin(x)}\, dx =$

Error in Midpoint Rule approximation:

Error in Simpson's Rule approximation: