Exercises require both the trapezoidal rule and Simpson’s rule. They can be worked without calculator programs if such programs are not available, although they require more calculation than the other problems in this exercise set.

The difference between the true value of an integral and the value given by the trapezoidal rule or Simpson’s rule is known as the error. In numerical analysis, the error is studied to determine how large n must be for the error to be smaller than some specified amount. For both rules, the error is inversely proportional to a power of n, the number of subdivisions. In other words, the error is roughly k/n^p, where k is a constant that depends on the function and the interval, and p is a power that depends only on the method used. With a little experimentation, you can find out what the power p is for the trapezoidal rule and for Simpson’s rule.

(a) Find the exact value of ∫₀¹x⁴ dx.

(b) Approximate the integral in part (a) using the trapezoidal rule with n = 4, 8, 16, and 32. For each of these answers, find the absolute value of the error by subtracting the trapezoidal rule answer from the exact answer found in part (a).
(c) If the error is k/n^p, then the error times n p should be approximately a constant. Multiply the errors in part (b) times n^p for p = 1, 2, etc., until you find a power p yielding the same answer for all four values of n.
(d) Based on the results above, what happens to the error in the trapezoidal rule when the number of intervals is doubled?