- #1
DaalChawal
- 87
- 0
I'm having a problem solving this, My approach is solving $x_1$ as a variable and rest as constants first and then going on further. But it is getting too lengthy. Is there any short method?
Shortcuts for Solving Multiple Integrals: Is There a Faster Way?
- MHB
- Thread starter DaalChawal
- Start date
In summary, a multiple integral is a mathematical concept used to calculate the area under a multivariable function over a given region in space. It is an extension of the single integral and is represented by the number of dimensions in which the calculation is being performed. This includes double integrals for two dimensions and triple integrals for three dimensions. Multiple integrals are commonly used in scientific research to calculate important quantities and evaluate complex functions in multidimensional space. The limits of integration for a multiple integral are determined by the boundaries of the surface or volume being integrated over and can be solved using numerical methods such as the Monte Carlo method or Simpson's rule.
I'm having a problem solving this, My approach is solving $x_1$ as a variable and rest as constants first and then going on further. But it is getting too lengthy. Is there any short method?
- #2
Prove It
Gold Member
MHB
- 1,465
- 24
Would it help writing the integrand as $\displaystyle \begin{align*} 1 - \frac{2\,x_5}{x_1+x_2+x_3+x_4+x_5} \end{align*}$?
- #3
DaalChawal
- 87
- 0
Yes, it does help but after the first two steps log comes, and then using by parts it becomes quite lengthy. I was looking for a short approach so that this question can be solved in 2-3 minutes. Btw thanks for your help 
1. What is the concept of "doubt" in multiple integrals?
Doubt in multiple integrals refers to uncertainty or lack of confidence in the solution or result obtained from solving a multiple integral. It can arise due to various reasons such as errors in calculations, limitations of the integration method used, or the complexity of the integrand.
2. How can one minimize doubt in multiple integrals?
To minimize doubt in multiple integrals, it is important to use reliable and accurate integration methods, double-check calculations, and use appropriate approximations if needed. It is also useful to have a good understanding of the integrand and the problem at hand.
3. Can doubt in multiple integrals affect the final result?
Yes, doubt in multiple integrals can significantly affect the final result. Even small errors or uncertainties in the integral can lead to a completely different result. This is especially true for complex integrands or high-dimensional integrals.
4. How can one determine if the result obtained from a multiple integral is reliable?
One way to determine the reliability of a result obtained from a multiple integral is to use different integration methods and compare the results. If they are consistent, then the result can be considered reliable. Additionally, checking the units and dimensions of the result can also help determine its validity.
5. What are some common sources of doubt in multiple integrals?
Some common sources of doubt in multiple integrals include errors in calculations, limitations of the integration method used, improper selection of variables or limits, and the presence of singularities or discontinuities in the integrand. It can also arise due to the complexity or ambiguity of the problem being solved.
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