Solving erf(1.00) with Trapezoid Rule (n=6)

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In summary: Add the areas together and you have an approximation for the definite integral. In summary, to approximate the value of erf(1.00) using the trapezoid rule with n=6, you need to construct 6 trapeziums with a width of 1/6 and calculate the areas using the function e^{-x^2} as the height at each point. Then, add the areas together to get an approximation for the definite integral.
  • #1
http://en.wikipedia.org/wiki/Error_function

If you have erf(1.00) and are asked to solve for the approximate value by using the trapezoid rule with n=6, how would you go about doing so?

Since the function is erf(x) the 'x' goes into the limits of integration but a 't' is used as a variable in the actula function. How do you know what to use for 'u'. Do you simply treat the function as an integral from 0 to 1 and evauluate 'u' from 0 to 1 as well?

thanks for any help?
 
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  • #2
Basically the question is asking to approximate the definite integral:

[tex]\frac{2}{\sqrt{\pi}}\int_0^1 e^{-x^2} dx[/tex]

They want you to do it with the trapezoidal rule. Basically construct 6 trapeziums, each with a width of 1/6. The height of the left side of the trapezium should be the same as [itex]e^{-x^2}[/itex] at that point, and the height of the right side should do the same for its point.

Eg The First trapezium Will have a base from 0 to 1/6. At 0, [itex]e^{-x^2}=1[/itex]. So the height at that point should be 1. At the other side of the trapezium, 1/6, sub in 1/6 into [itex]e^{-x^2}[/itex], and that's the height at that point. You have the two heights of the trapezium, and a base length, now you can find the area.

Repeat for the other 5 trapeziums.
 
  • #3


To solve for erf(1.00) using the trapezoid rule with n=6, we must first understand the trapezoid rule and how it can be applied to this problem. The trapezoid rule is a numerical integration method that approximates the area under a curve by dividing the region into trapezoids and summing their areas. In this case, we are trying to approximate the area under the erf(x) curve from 0 to 1, which will give us the value of erf(1.00).

To use the trapezoid rule, we need to determine the limits of integration, the number of trapezoids (n), and the function itself. In this case, the limits of integration are from 0 to 1, n=6, and the function is erf(x). However, as you mentioned, the function uses 't' as the variable, so we need to make a substitution to match the limits of integration. We can do this by using the following substitution: t = (2*x - 1)/√(π).

Now, our limits of integration become t = 0 when x = 0 and t = 1 when x = 1. We can then rewrite the function as erf(x) = √(π)/2 * ∫0^1 e^(-t^2) dt.

Using the trapezoid rule, we can approximate the area under the curve by dividing the region into 6 trapezoids and calculating their areas. The formula for the trapezoid rule is A = (1/2)(b-a)(f(a) + f(b)), where a and b are the limits of integration and f(a) and f(b) are the function values at those points.

In this case, our limits of integration are a = 0 and b = 1, and we can calculate the function values at those points by plugging them into our substituted function: f(a) = e^(-0^2) = 1 and f(b) = e^(-1^2) = 0.3679.

Plugging these values into the trapezoid rule formula, we get A = (1/2)(1-0)(1+0.3679) = 0.6839. This means that the approximate value of erf(1.00) using
 

What is the Trapezoid Rule for solving erf(1.00)?

The Trapezoid Rule is a numerical method for approximating the definite integral of a function. In the context of solving erf(1.00), it is used to approximate the area under the curve of the error function.

What is the value of n in the Trapezoid Rule for solving erf(1.00)?

n refers to the number of trapezoids used in the approximation. In this case, n=6 means that the area under the curve will be divided into 6 trapezoids to calculate the approximate value of erf(1.00).

How accurate is the Trapezoid Rule for solving erf(1.00)?

The accuracy of the Trapezoid Rule depends on the value of n. Generally, the larger the value of n, the more accurate the approximation will be. However, it is important to note that the Trapezoid Rule is still an approximation and may not give the exact value of erf(1.00).

How is the Trapezoid Rule used to solve erf(1.00)?

The Trapezoid Rule involves dividing the area under the curve of erf(1.00) into trapezoids and calculating the area of each trapezoid. Then, the areas are added together to approximate the total area under the curve and thus, the value of erf(1.00).

Can the Trapezoid Rule be used to solve other functions besides erf(1.00)?

Yes, the Trapezoid Rule can be used to approximate the integral of any function. However, the accuracy of the approximation may vary depending on the function and the value of n used.

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