Change integration limits for cylindrical to cartesian coord

In summary, the conversation discusses changing the integration limits of an integral from cylindrical to cartesian coordinates. The suggested method is to use the equations r=√(x^2+y^2) and rdr=dxdy, and to indicate an integration order for x and y. However, it is pointed out that it is not possible to rewrite a one-variable integral into a two-variable integral in this way. The correct equation is dx dy = r dr dφ.
  • #1
MCB277
6
0

Homework Statement


I want to change the integration limits of an integral in cylindrical to cartesian coordinates. For example the integral of function f(r) evaluated between b and R: ∫ f(r)dr for r=b and r=R (there is no angular dependence).

For write de function in cartesian coordinates, use r=√(x^2+y^2) and rdr=dxdy, then, I should indicate an integration order for x and y.

Homework Equations


r=√(x^2+y^2)
∫ f(x,y)dx dy for x=? and y=?

The Attempt at a Solution



If I integrate in x first, de limit of integration should be x=-√(b^2-y^2) and x=-√(R^2-y^2), but for "y", what happens?.

Thanks
 
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  • #2
MCB277 said:
dr=dxdy
This is not correct. You cannot rewrite a one-variable integral into a two-variable integral like that. What you are looking for is ##dx\,dy = r\, dr\, d\varphi##.
 

Related to Change integration limits for cylindrical to cartesian coord

What is the process for changing integration limits from cylindrical to cartesian coordinates?

To change integration limits from cylindrical to cartesian coordinates, you must first determine the relationship between the two coordinate systems. For example, in cylindrical coordinates, the radius is represented by r and the angle is represented by θ, while in cartesian coordinates, they are represented by x and y. Then, use this relationship to convert the integration limits accordingly.

Why would you need to change integration limits from cylindrical to cartesian coordinates?

You may need to change integration limits from cylindrical to cartesian coordinates when solving a problem that involves both coordinate systems. This could be because the given equation or region is better suited for one coordinate system over the other, or it could be a requirement for a specific calculation or application.

Can you provide an example of changing integration limits from cylindrical to cartesian coordinates?

Yes, for example, let's say you have an equation in cylindrical coordinates, such as f(r,θ) = r2 + 3cos(θ), and you want to find the volume under the curve in the region r ≤ 2 and 0 ≤ θ ≤ π/2. To change the integration limits to cartesian coordinates, we can use the relationship r2 = x2 + y2 and cos(θ) = x/r. Therefore, the new integration limits would be 0 ≤ y ≤ √(4-x2) and 0 ≤ x ≤ 2.

Are there any special considerations when changing integration limits from cylindrical to cartesian coordinates?

Yes, there are a few things to keep in mind. First, make sure to use the correct relationship between the two coordinate systems. Second, pay attention to the orientation of the region and adjust the integration limits accordingly. Third, be aware of any singularities or discontinuities that may arise when converting the integration limits.

Are there any tools or techniques to help with changing integration limits from cylindrical to cartesian coordinates?

Yes, there are various online calculators and software programs that can help with converting integration limits between coordinate systems. Additionally, it is helpful to have a good understanding of the properties and relationships of different coordinate systems, as well as practice and familiarity with solving problems involving integration in multiple coordinate systems.

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