Approximating integral of x^4 e^-x

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In summary, the speaker is trying to approximate an integral of the form \int_0^{r_0} r^4 e^{-r/a} dr, using the expansion of the exponential and integrating the first few terms. However, they noticed that the error gets bigger as r0/a gets smaller. They suspect it may be due to numerical error and mention that there are also 1/r^5 and 1/r^6 terms in the taylor approximation. They also mention another integral, \int^{1} _{0} x^{m} e^{-bx} dx, which has an exact answer of \frac{m!}{b^{m+1}} \left[ 1 - e^{-b} \
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I'm trying to approximate an integral of the form:

[tex]\int_0^{r_0} r^4 e^{-r/a} dr [/tex]

where r0<<a. I figured I could write out the first few terms of the expansion of the exponential and integrate that, since the extra terms would quickly become negligible:

[tex] \int_0^{r_0} r^4 (1-\frac{r}{a}) dr [/tex]

However, when I calculate this answer and compare it to the exact answer from mathematica, the error gets bigger the smaller r0/a gets. I can't figure out why. Specifically, with a few extra terms with the same form, but different powers of r, and r0/a of the order 10^-5, my approximated integral comes out to be about 10^-10 while the exact value is about 0.02. Could this be numerical error? I told it to algebraically come up with the ratio of the approximation and the exact integral, and there were 1/r^5 and 1/r^6 terms, but I can't figure out where they're coming from. Even as I add more terms to the taylor approximation, the answer barely changes.
 
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  • #2
express the integral in the form
[tex] \int^{1} _{0} x^{m} e^{-bx} dx [/tex] the exact answer is

[tex] \frac{m!}{b^{m+1}} \left[ 1 - e^{-b} \sum^{m} _{r = 0} \frac{b^{r}}{r!} \right] [/tex]
 
  • #3
Have you considered that, after you've approximated the integrand, you made a mistake in your numerical computation of the integral?
 

Related to Approximating integral of x^4 e^-x

What is the purpose of approximating the integral of x^4 e^-x?

The purpose of approximating the integral of x^4 e^-x is to find an estimate of the total area under the curve of the function x^4 e^-x. This can be useful in various scientific and mathematical applications, such as calculating probabilities or solving differential equations.

What is the general method for approximating integrals?

The general method for approximating integrals is to divide the interval of integration into smaller subintervals and use a numerical method, such as the trapezoidal rule or Simpson's rule, to calculate the area under the curve for each subinterval. The sum of these areas gives an approximation of the integral.

Why is it difficult to find an exact solution for the integral of x^4 e^-x?

It is difficult to find an exact solution for the integral of x^4 e^-x because this function does not have an antiderivative that can be expressed in terms of elementary functions. As a result, numerical methods must be used to approximate the integral.

How does the accuracy of the approximation change with the number of subintervals?

The accuracy of the approximation generally increases as the number of subintervals used in the numerical method increases. This is because the smaller the subintervals, the closer the approximation is to the actual area under the curve. However, using too many subintervals can also lead to numerical errors and may not improve the accuracy significantly.

What are some potential sources of error in approximating integrals?

There are several potential sources of error in approximating integrals, including rounding errors in calculations, using too few subintervals, and using a numerical method that is not suitable for the function being integrated. It is important to carefully choose the appropriate method and number of subintervals to minimize these errors.

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