How to perform multivariable numerical integration?

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SUMMARY

This discussion addresses the process of numerically integrating multivariable functions, specifically using the trapezoidal rule for functions like P = (x² + 4xy). The user initially expressed confusion about integrating multiple variables but later clarified that for two variables, the reference "Numerical Quadrature and Cubature" by H. Engels (1980) provides guidance. For functions with more than two variables, Monte Carlo integration is recommended as an effective method.

PREREQUISITES
  • Understanding of numerical integration techniques, specifically the trapezoidal rule.
  • Familiarity with multivariable calculus concepts.
  • Knowledge of Monte Carlo integration methods.
  • Access to "Numerical Quadrature and Cubature" by H. Engels for theoretical insights.
NEXT STEPS
  • Research the implementation of the trapezoidal rule for multivariable functions.
  • Study Monte Carlo integration techniques for high-dimensional numerical integration.
  • Explore additional resources on numerical methods for multivariable calculus.
  • Review case studies or examples of numerical integration in scientific computing.
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Mathematicians, engineers, and data scientists who require techniques for numerical integration of multivariable functions, particularly those working with complex models in simulations or computational analysis.

Topher925
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I've got a problem where I need to numerically integrate a multivariable function but I'm not sure how to do this. I'm more than familiar with how to numerically integrate a single variable function numerically but I have no clue how to do this for a multivariable function. For example let's say I have the function

P = (x2 + 4xy)

and I need to integrate this numerically between some definite closed bounds of x, y, and z using the trapezoidal rule. How would one go about doing this? Would I simply perform the integration on each variable and then plug that solution into the respective variable for that function for each integral I evaluate? I have two books on numerical methods and none of them mention anything about doing this. Is it even possible?
 
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Nevermind, I figured it out. Just had a bit of a stupid moment.
 
For the record: with 2 variables, see H. Engels, "Numerical Quadrature and Cubature", Academic, 1980. For many variables, use Monte Carlo integration.
 

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