Require both the trapezoidal rule and Simpson’s rule. They...

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. Repeat Exercise 15 using Simpson’s rule.

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.

Repeat Exercise 15 using Simpson’s rule.

Key Concepts

Error Analysis in Numerical Methods

Error analysis in numerical integration studies the difference between the true value of an integral and its numerical approximation. It involves understanding how the error decreases as the number of subdivisions increases, often characterized by a relationship where the error is proportional to k/n^p, with k being dependent on the function and p indicating the order of accuracy.

Simpson’s Rule

Simpson’s rule is a numerical integration technique that approximates the integral by using quadratic polynomials to model the function over subintervals. This method typically provides a more accurate approximation compared to the trapezoidal rule for the same number of subdivisions, particularly when the function behaves smoothly.

Order of Convergence

The order of convergence is a concept that describes how quickly the error term in a numerical method decreases as the number of subdivisions (or the step size) is refined. In the context of numerical integration, methods like the trapezoidal and Simpson’s rules exhibit different orders of convergence, meaning that the error decreases at different rates relative to n.

Numerical Integration

Numerical integration refers to the process of approximating the value of a definite integral when an exact analytical integration is difficult or impossible. This concept covers methods and algorithms that compute approximate values of integrals by summing function evaluations multiplied by weights determined by the method used.

Trapezoidal Rule

The trapezoidal rule is a numerical integration method that approximates the area under a curve by dividing it into trapezoids rather than rectangles. It uses linear interpolation between points on the function and obtains an approximation that improves with a larger number of subdivisions.

Transcript

00:01 Okay, so for this particular problem, we are asked to calculate the actual, the exact value of the integral from zero to one of x to the fourth dx.

00:12 So this is going to be, we're going to add one to the exponent and then multiply this by the reciprocal of that exponent, which will be one -fifth.

00:21 And we're looking at this from zero to one.

00:23 So this is going to be one -fifth or point two.

00:28 So this would be our exact value.

00:30 So now what it's asking us to do is to evaluate utilizing a calculator.

00:37 We're going to look at simpson's rule when n is 4, when n is 8, when n is 16, and, last but not least, when n is 32.

00:53 Okay? so what i want to do is i'm going to give the approximations that i found with the calculator.

00:59 So for the first one, i got 2 .2 .003 -225.

01:08 And if we evaluate the error, the formula for finding the error is going to be the exact value, which we found in part a to be .2, minus this long number that we just found using a simpson rule calculator.

01:25 So this is going to be .0 .000 .0 .0 .0.

01:34 3 -225.

01:41 And then for when n is 8, simpson rule is telling us that it's 0 .000 -200 -203.

01:50 So the error, again, is going to be that 0 .2 value we found an a minus this value.

01:59 So that's going to be 0 .0 -0 -0 -0 -2002 -2 -0 -2 -3.

02:09 And then moving on to where n is 16, the calculator tells me that it's 0 .2.

02:14 0 .001 -127.

02:20 So it's going to be 0 .2 minus this value...