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 $\frac{k}{n^p}$ 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. Complete parts (a) through (c) below.
(a) Find the exact value of $\int_0^1 3x^4 \, dx$.
$\int_0^1 3x^4 \, dx = 0.6$ (Type an integer or a decimal.)
(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).
Complete the table below.
n Trapezoidal Rule Absolute Value of the Error Approximation
4
8
16
32
(Round to six decimal places as needed.)
