An integral with variables

In summary, the conversation is about evaluating an integral with multiple variables and finding the value using different coordinate systems. The integral is originally in Cartesian coordinates but can be converted to polar or rotated to align with the (x,y,z,...) vector. The final result is a function of the length of the vector and the number of variables.
  • #1
forumfann
24
0
Could anyone help me evaluate the integral
[itex]
\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}|sx+ty|e^{-s^{2}/2}e^{-t^{2}/2}dsdt
[/itex], which should be a function of x and y?

By the way, this is not a homework problem.

Thanks
 
Last edited:
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  • #2
Well, make a shift to polar coordinates:
[tex]s=r\cos\theta,t=r\sin\theta[/tex]
[tex]x=R\cos\phi,y=R\sin\phi[/tex]

Thus, your integral becomes:
[tex]R\int_{0}^{\infty}\int_{0}^{2\pi}|\cos(\theta-\phi)|r^{2}e^{-\frac{r^{2}}{2}}d\theta{d}r[/tex]
 
  • #3
Thanks a lot for arildno's help. So I am able to get the value of the integral with 2 variable now.

But then how about 3 variables, i.e.
[tex]
\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}|rx+sy+tz|e^{-r^{2}/2}e^{-s^{2}/2}e^{-t^{2}/2}drdsdt ?
[/tex]
 
  • #4
Spherical coordinates, perchance??
 
  • #5
Instead of polar or spherical coordinates, you can also rotate your axis in the (r,s,t,...) space so that one of your axis becomes aligned with the (x,y,z,...) vector. The expression in the exponential is invariant under ratations, so what happens is that the integration becomes:


Integral dt1 dt2...dtn |y t1| exp(-t1^2/2)exp(-t2^2/2)...
exp(-tn^2/2) =

2|y| (2pi)^[(n-1)/2]

where, of course, y = the length of your (x,y,z,...) vector
 
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1. What is an integral with variables?

An integral with variables is a mathematical expression that represents the accumulation of a quantity over an interval. It is used to find the total value of a function or rate of change over a given range of values.

2. How is an integral with variables different from a basic integral?

An integral with variables differs from a basic integral in that the upper and lower limits of the integration are not fixed constants, but instead are represented by variables. This allows for a more general and flexible representation of the integration process.

3. What is the significance of using variables in an integral?

Using variables in an integral allows for more complex and dynamic calculations, as the integration can be performed over a range of values rather than just a fixed set of values. This is especially useful in real-world applications where the limits of integration may vary.

4. How do you solve an integral with variables?

To solve an integral with variables, you must first determine the limits of integration by finding the values of the variables that correspond to the start and end points of the interval. Then, you can use various integration techniques, such as substitution or integration by parts, to find the definite integral of the function.

5. What are some common applications of integrals with variables?

Integrals with variables have many real-world applications, including calculating areas and volumes of irregular shapes, determining rates of change in physics and economics, and solving optimization problems in engineering and math.

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