Can Someone Help Me with This Integral?

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In summary, the conversation discusses a question about a specific integral involving vectors and an arbitrary vector, with a circular area as the domain of integration. One person suggests a solution involving symmetry, but another points out that it is not zero according to Wolfram Alpha due to ignoring the original poster's notation and the domain of integration.
  • #1
hiyok
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Hi,

I'm stumbled on the following integral,

[itex]\int d\vec{q} e^{i\vec{q}\vec{r}} \cos(2\theta)[/itex],

where [itex] \theta[/itex] denotes the angle of the vector [itex] \vec{q} [/itex] and the [itex] \vec{r}[/itex] is an arbitrary vector. The integral domain is a circular area of radius [itex] D [/itex].

Could anyone help me out ?

Many thanks.

hiyok
 
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  • #2
hiyok said:
Hi,

I'm stumbled on the following integral,

[itex]\int d\vec{q} e^{i\vec{q}\vec{r}} \cos(2\theta)[/itex],

where [itex] \theta[/itex] denotes the angle of the vector [itex] \vec{q} [/itex] and the [itex] \vec{r}[/itex] is an arbitrary vector. The integral domain is a circular area of radius [itex] D [/itex].

Could anyone help me out ?

Many thanks.

hiyok

Try cos(2θ) =(exp(2iθ)+exp(-2iθ))/2
 
  • #3
Thanks for your response.

I have worked it out. It amounts to zero by symmetry.
 
  • #5
amind said:
> I have worked it out. It amounts to zero by symmetry.
Although , I couldn't solve but it is not zero according to wolfram alpha.
see : http://www.wolframalpha.com/input/?i=∫+(e^(i*q*r)+cos(2θ))dq
If you're going to ignore both the original poster's notation (##\theta## has a specific meaning, and it's a vector integral) and the domain of integration, you can't expect to just type the integral into Wolfram Alpha and expect to get the right result.
 
  • #6
eigenperson said:
If you're going to ignore both the original poster's notation (##\theta## has a specific meaning, and it's a vector integral) and the domain of integration, you can't expect to just type the integral into Wolfram Alpha and expect to get the right result.
I'm sorry for that , I'm not great in calculus and I probably skimmed through the posts.
:(
 

1. What is an integral and why is it important?

An integral is a mathematical concept that represents the area under a curve. It is important because it allows us to find the exact value of quantities such as displacement, velocity, and acceleration, which are essential in many fields of science such as physics, engineering, and economics.

2. How do I know which method to use to solve an integral?

The method you use to solve an integral depends on the form of the integral. Some common methods include u-substitution, integration by parts, and trigonometric substitution. It is important to first identify the form of the integral and then choose the appropriate method to solve it.

3. Can I use a calculator to solve integrals?

Yes, there are many online and offline calculators available that can solve integrals. However, it is important to note that these calculators may not always give the most accurate or complete solution, so it is recommended to also know how to solve integrals by hand.

4. Are there any tips or tricks for solving integrals more quickly?

Yes, there are a few tips and tricks that can help you solve integrals more efficiently. These include using u-substitution, recognizing common patterns, and breaking up the integral into smaller, more manageable parts. Practice and familiarity with different types of integrals can also help speed up the process.

5. What are some real-life applications of integrals?

Integrals have a wide range of applications in science and engineering. They are used in physics to calculate work, energy, and power, in economics to determine the optimal production and consumption levels, and in biology to model populations and growth rates. They are also used in various engineering fields to design and optimize systems and processes.

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