Exercises 15-8 require both the trapezoidal rule and...

Exercises 15-8 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. Error Analysis 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. $$\begin{array}{l}{\text { (a) Find the exact value of } \int_{0}^{1} x^{4} d x} \\ {\text { (b) Approximate the integral in part (a) using the trapezoidal rule }} \\ {\text { with } n=4,8,16, \text { and } 32 . \text { For each of these answers, find }} \\ {\text { the absolute value of the error by subtracting the trapezoidal }} \\ {\text { rule answer from the exact answer found in part (a). }}\end{array}$$ $$\begin{array}{l}{\text { (c) If the error is } k / n^{p} \text { , then the error times } n^{p} \text { should be ap- }} \\ {\text { proximately a constant. Multiply the errors in part (b) times }} \\ {n^{p} \text { for } p=1,2, \text { etc., until you find a power } p \text { yielding the }} \\ {\text { same answer for all four values of } n .}\end{array}$$

Exercises 15-8 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.
Error Analysis 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.
$$\begin{array}{l}{\text { (a) Find the exact value of } \int_{0}^{1} x^{4} d x} \\ {\text { (b) Approximate the integral in part (a) using the trapezoidal rule }} \\ {\text { with } n=4,8,16, \text { and } 32 . \text { For each of these answers, find }} \\ {\text { the absolute value of the error by subtracting the trapezoidal }} \\ {\text { rule answer from the exact answer found in part (a). }}\end{array}$$
$$\begin{array}{l}{\text { (c) If the error is } k / n^{p} \text { , then the error times } n^{p} \text { should be ap- }} \\ {\text { proximately a constant. Multiply the errors in part (b) times }} \\ {n^{p} \text { for } p=1,2, \text { etc., until you find a power } p \text { yielding the }} \\ {\text { same answer for all four values of } n .}\end{array}$$

Key Concepts

Numerical Integration

Numerical integration refers to a collection of techniques used to approximate the value of definite integrals, especially when an antiderivative of the function is difficult or impossible to obtain analytically. These methods use discretization of the integration interval and summing function values at specified points to approximate the total area under a curve.

Trapezoidal Rule

The trapezoidal rule is a numerical integration technique that approximates the area under a curve by dividing the integration interval into subintervals and estimating the area using trapezoids rather than the exact curve. The approximation accuracy increases as the number of subdivisions (n) increases, with the error generally decreasing in proportion to 1/n² under standard smoothness conditions.

Simpson's Rule

Simpson's rule is another numerical integration method that approximates an integral by fitting quadratic polynomials (parabolic arcs) between pairs of subintervals. This method typically yields a more accurate approximation than the trapezoidal rule for smooth functions, as its error tends to decrease at a faster rate, often proportional to 1/n?, given sufficient smoothness of the function.

Error Analysis

Error analysis in numerical integration is the study of how the approximation error—the difference between the true value of the integral and the numerical approximation—behaves as the parameters of the numerical method change. It involves determining how many subdivisions are needed to achieve a desired accuracy, often by expressing error as a constant times a power of the number of subdivisions, i.e., k/n?.

Order of Convergence

The order of convergence refers to the exponent p in the error estimate (error ~ k/n?), which indicates the rate at which the error decreases as the number of subdivisions increases. A higher order of convergence means that the error diminishes faster with increasing n, thus allowing for higher accuracy with fewer subdivisions. Determining the order of convergence experimentally helps in understanding the efficiency and performance of different numerical integration methods.

Transcript

00:01 So for this question, we're going to talk about error analysis for the trapezoidal rule.

00:06 So before we do that, we're asked to find the exact value of the integral first.

00:11 So let us do that.

00:15 So part a, we want to find the exact value of the integral from 0 to 1 of x to the power of 4dx.

00:24 So that is going to be 1 over 5, x the power of 5, and that's going to be evaluated from 0 to 1.

00:33 And that's just equal to 1 over 5 or 0 .2.

00:40 Now part b is asking us to use the trapezoidal rule, and we need to repeat that for different values of n.

00:48 So we need to do it for n equals 4, 8, 16, and 32.

00:53 So i won't be writing all the numbers down because i don't have enough space to write everything down.

01:00 But what you need to do is you need to, so for example, for n equals 4, you need to find your delta x.

01:10 So for n equals 4, our delta x is going to be b minus a over n, so that's 1 over 4.

01:18 So once you have that, you need to write down all your x -is.

01:22 So my x -is, i'll start at 0 because that's my a, and i'm going to go up by a quarter.

01:30 So that's 0 .25, 0 .5, 0 .75, and 1.

01:37 And then once you find those, you need to plug into your 4 .25 ,000, and 1.

01:40 Function x of power 4 and find what your y value is so that'll get us 0 .0 .003906 0 .0 .0 625 0 .3164 and 1.

02:03 And once you have all your values i have written the trapezoidal rule up here in red.

02:09 So you're going to plug these values into this formula.

02:14 And once you do that, you'll get integral from 0 to 1 of x of power 4 x, that is going to be approximately 0 .2207.

02:32 So once you've done that, now you need to repeat the process.

02:36 So you're going to do it for n equals 8, n equals 16, and n equals 32.

02:45 So like i said earlier, i'm not going to write down all these values.

02:50 I have done it already in a separate program.

02:56 So i have the values.

02:58 And you can do the calculations yourself and just double check.

03:03 But for n equals 8, we're going to get, for n equals 8, we will get the value is approximately 0 .205.

03:22 For n equals 16, it's going to be 0 .2013.

03:31 And lastly, for n equals 32, we'll get 0 .203.

03:43 So with these four values, now you're asked to calculate what is the absolute value of the error by subtracting your approximation with the exact answer for.

03:55 From part a.

03:56 So that part's easy.

03:58 All you're doing is you're taking this value and you're subtracting the exact value which is 0 .2 from that.

04:06 So which means our errors we're going to get for n equals 4, that's going to be 0 .0207.

04:28 For an n equals 8, our error is going to be 0 .00 .7...

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