Finding new region for double integral

In summary, the conversation is about a double integral with a region of integration given by ##R = \{(u,v) : 0 \le u < \infty, ~0 \le v < \infty) \}##. The individual is using a change of variables ##u=zt## and ##v = z(1-t)## and is unsure how to find the new region of integration in terms of the zt-coordinate system. They are advised to draw a sketch and express u=0 and v=0 in the new coordinates. It is also suggested to take the process step by step and consider additional options for u and v. After making the necessary replacements, the individual is able to determine that if u
  • #1
Mr Davis 97
1,462
44
I have a double integral where the region of integration is ##R = \{(u,v) : 0 \le u < \infty, ~0 \le v < \infty) \}##. I am doing the change of variables ##u=zt## and ##v = z(1-t)##. I am a bit rusty on calc III material, so how would I find the new region of integration, in terms of the zt-coordinate system?
 
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  • #2
Did you draw a sketch?
Did you express u=0 and v=0 in the new coordinates?
 
  • #3
mfb said:
Did you draw a sketch?
Did you express u=0 and v=0 in the new coordinates?
Will I know that in uv coordinates the region is the first quadrant.

If u=0, then z=v.
If v=0, then z=u.

Does this help me in any way?
 
  • #4
Mr Davis 97 said:
If u=0, then z=v.
If v=0, then z=u.
These are not the only options, and you are trying to do two steps at the same time. Do it step by step.
 
  • #5
mfb said:
These are not the only options, and you are trying to do two steps at the same time. Do it step by step.
If ##x=0,y \ne 0##, then ##z=y,t=0##
If ##y=0,x \ne 0##, then ##z=x, t=1##.

Is this correct?
 
  • #7
mfb said:
What are x and y?
Sorry, replace x with u and y with v.
 
  • #8
And if u=v=0 then z=0.
That looks good and it should tell you what to integrate over.
 

Related to Finding new region for double integral

1. How do you determine the limits of integration for a double integral?

The limits of integration for a double integral are determined by the region over which the integral is being evaluated. This region can be defined by the intersection of two curves or by a combination of lines, curves, and points. It is important to carefully analyze the region and identify the equations that describe its boundaries in order to correctly set up the limits of integration.

2. What is the difference between a rectangular and a polar region for a double integral?

A rectangular region is defined by vertical and horizontal lines, while a polar region is defined by a single curve and a range of angles. In a rectangular region, the limits of integration are constant, while in a polar region, the limits will vary depending on the angle.

3. Can the region for a double integral be defined by inequalities?

Yes, the region for a double integral can be defined by inequalities. This is commonly seen when the region is defined by a combination of curves and lines. In this case, the inequalities are used to specify the boundaries of the region.

4. How do you determine the type of coordinate system to use for a double integral?

The type of coordinate system to use for a double integral is determined by the shape and symmetry of the region. If the region has rectangular symmetry, it is best to use rectangular coordinates. If the region has polar symmetry, polar coordinates should be used. If the region has a combination of both rectangular and polar symmetry, it may be necessary to use a combination of both coordinate systems.

5. What is the importance of identifying the correct region for a double integral?

Identifying the correct region for a double integral is crucial because it determines the limits of integration and the type of coordinate system that should be used. Incorrectly identifying the region can lead to incorrect solutions and a lack of understanding of the problem at hand. It is important to carefully analyze the region in order to accurately set up and solve the double integral.

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