How do I approximate the integral

In summary, the conversation discusses how to approximate the integral of e^{-x^2} over some interval by using a series expression and antidifferentiating the series. It also mentions using a finite number of terms to estimate the antiderivative and the availability of a tabulated error function for the calculation.
  • #1
Icebreaker
How do I approximate the intergral of [tex]e^{-x^2}[/tex] over some interval?
 
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  • #2
[tex]e^{r} = \sum_{n=0}^\infty \frac{r^n}{n!}[/tex]

Substitute r = -(x^2).

[tex]e^{-x^2} = \sum_{n=0}^\infty \frac{(-1)^{n}x^{2n}}{n!}[/tex]

Antidifferentiate series over x.


[tex]\int {e^{-x^2}}dx = \sum_{n=0}^\infty \frac{(-1)^{n}x^{2n+1}}{(2n+1)n!} = \sum_{n=0}^\infty \frac{(-1)^{n}x^{2n+1}}{(2n+1)n!}[/tex]

You can use the last expression with some finite upper value. In other words,

[tex]\sum_{i=0}^k \frac{(-1)^{i}x^{2i+1}}{(2i+1)i!}[/tex]

will yield an estimate of the desired antiderivative, with (k+1) being the number of terms involved.

[Edit: Sorry! Initially, I differentiated the series, instead of antidifferentiating.]
 
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  • #3
U differentiated the series,u should have integrated it.

[tex]\int e^{-x^{2}} \ dx=\frac{\sqrt{\pi}}{2}\mbox{erf}\left(x\right) + C [/tex]

and the error function is tabulated...

Daniel.
 
  • #4
Yeah, I realized that this morning.

Sorry, it was kind of late.
 
  • #5
Thanks to both
 

Related to How do I approximate the integral

1. What is the purpose of approximating an integral?

Approximating an integral allows us to estimate the area under a curve when we do not have an exact mathematical formula to find the exact value. This is useful in real-world applications where we need to find an approximate solution quickly.

2. How do I choose the appropriate method for approximating an integral?

The method for approximating an integral depends on the type of function and the accuracy needed. Some common methods include the trapezoidal rule, Simpson's rule, and the midpoint rule. It is important to understand the strengths and limitations of each method before choosing the most appropriate one.

3. Can I use a computer program to approximate an integral?

Yes, there are many numerical integration methods available in computer programs such as MATLAB or Python. These programs use algorithms to calculate the approximate value of the integral based on the chosen method and input parameters.

4. What is the difference between an exact and an approximate integral?

An exact integral is calculated using a mathematical formula, while an approximate integral is calculated using numerical methods. An exact integral gives the exact value of the area under the curve, while an approximate integral gives an estimate that may have some level of error.

5. How do I know if my approximation is accurate enough?

The accuracy of an approximation can be determined by comparing it to the exact value, if available. Additionally, we can also use error estimation formulas to determine the maximum possible error for a given method. It is important to consider the level of accuracy needed for the specific application when evaluating the accuracy of an approximation.

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