In this problem you will use the midpoint rule, the trapezoid rule, and Simpson's rule to estimate the value of the integral $\displaystyle I = \int_{1}^{9} \cos(x/5) \,dx$.
Since an antiderivative of the integrand can be found, we would not usually use approximate integration for this integral. But this fact allows us to compute the errors of the various approximation methods.

The exact value of this integral is $I =$ . (If you use a calculator, be sure you're in radian mode. You may want to store this result as $I$ for use in later parts of the problem. You can round the answer to 4 significant figures before typing into webwork, but be sure to store the unrounded result.)

The approximation to the integral using the midpoint rule with 2 subdivisions is $M_2 =$ . (You may want to store this as $M$ in your calculator.)
The signed error of this approximation is $M_2 - I =$ .

The approximation to the integral using the trapezoid rule with 2 subdivisions is $T_2 =$ . (You may want to store this as $T$ in your calculator.)
The signed error of this approximation is $T_2 - I =$ .

Notice that the signed error of the trapezoid rule is about twice the absolute value of, and has the opposite sign of, the signed error of the midpoint rule. Simpson's rule is the weighted average of the midpoint rule and the trapezoid rule, where the midpoint rule has twice the weight of the trapezoid rule. In general, $S_{2n} = \frac23 M_n + \frac13 T_n$. For example, $S_4 = \frac23 M_2 + \frac13 T_2$.

The approximation to the integral using Simpson's rule with 4 subdivisions is $S_4 =$ . (You may want to store this as $S$ in your calculator.)
The signed error of this approximation is $S_4 - I =$ .

Watch for roundoff error. You will need to have very accurate values for $S_4$ and $I$ in order to get the error $S_4 - I$ accurate to four significant figures. Simpson's rule is very accurate!