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

[elemental_d]

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?

 

[Gib Z]

Basically the question is asking to approximate the definite integral:

$$\frac{2}{\sqrt{\pi}}\int_0^1 e^{-x^2} dx$$

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 $e^{-x^2}$ 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, $e^{-x^2}=1$. So the height at that point should be 1. At the other side of the trapezium, 1/6, sub in 1/6 into $e^{-x^2}$, 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.

 

[tacman]

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