Domain for Polar Coordinate Part 2

In summary, the conversation discusses the problem of finding the integral of a function f(x,y) = e^{x^2+y^2} over a region where x^{2} + y^{2} ≤ R. The attempted solution involves using polar coordinates and performing a change of variables, but there is an error in the lower limit for r. The correct solution is given as \frac{1}{2}f^{2pi}_{0}f^{R}_{0}e^{u}*du.
  • #1
DrunkApple
111
0

Homework Statement


f(x,y) = [itex]e^{x^2+y^2}[/itex]
[itex]x^{2}[/itex] + [itex]y^{2}[/itex] ≤ R

Homework Equations





The Attempt at a Solution


I believe this is a circle.

[itex]f^{2pi}_{0}[/itex][itex]^{sqrt(R)}_{-sqrt(R)}[/itex][itex]e^{x^2+y^2}[/itex]*r*dr*dθ

= [itex]f^{2pi}_{0}[/itex][itex]f^{sqrt(R)}_{-sqrt(R)}[/itex][itex]e^{r^2}[/itex]*r*dr*dθ

after u substitution...

= [itex]\frac{1}{2}[/itex][itex]f^{2pi}_{0}[/itex][itex]f^{R}_{R}[/itex][itex]e^{u}[/itex]*du

Does this makes sense?

Of course not... r domain are same...
 
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  • #2
Polar coordinates are defined on the domain r>=0 and 0<=theta<2*pi. r is nonnegative. Your lower limit for r should be 0.
 
  • #3
ohhh so r can never be negative so they are always equal to or bigger than 0?
ok I got it.
Thank u

So the correct one is

= [itex]\frac{1}{2}[/itex][itex]f^{2pi}_{0}[/itex][itex]f^{R}_{0}[/itex][itex]e^{u}[/itex]*du
 
  • #4
DrunkApple said:
ohhh so r can never be negative so they are always equal to or bigger than 0?
ok I got it.
Thank u

So the correct one is

= [itex]\frac{1}{2}[/itex][itex]f^{2pi}_{0}[/itex][itex]f^{R}_{0}[/itex][itex]e^{u}[/itex]*du

Use \int for a [itex]\int[/itex] in LaTeX.
 

FAQ: Domain for Polar Coordinate Part 2

What is the purpose of using polar coordinates?

Polar coordinates are used to represent points in a two-dimensional plane. They are particularly useful for describing circular or symmetrical shapes, as they use a distance from the origin (known as the radius) and an angle from a reference axis (known as the polar angle) to specify a point. This can make calculations and graphing of these shapes easier and more intuitive.

How do you convert between polar and Cartesian coordinates?

To convert from polar coordinates to Cartesian coordinates, you can use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
where r is the radius and theta is the polar angle. To convert from Cartesian coordinates to polar coordinates, you can use the following formulas:
r = square root of (x^2 + y^2)
theta = arctan(y/x)
Note that the polar angle may need to be adjusted depending on the quadrant of the point.

What is the difference between polar coordinates and rectangular coordinates?

Polar coordinates use a distance from the origin and an angle from a reference axis to specify a point, while rectangular coordinates use x and y coordinates to specify a point. This means that polar coordinates are more useful for describing circular or symmetrical shapes, while rectangular coordinates are better for describing more complex shapes or for more precise measurements.

Can polar coordinates be used in three-dimensional space?

While polar coordinates are typically used in two-dimensional space, they can also be extended to three-dimensional space. In this case, a third coordinate, known as the polar elevation angle, is added to specify the point's location along the z-axis. However, rectangular coordinates are still more commonly used in three-dimensional space as they are better suited for describing complex shapes and objects.

What are some common uses of polar coordinates in science and engineering?

Polar coordinates have various applications in science and engineering. They are commonly used in fields such as physics, astronomy, and engineering to describe circular or rotational motion, electromagnetic fields, and other symmetrical phenomena. They are also commonly used in mapping and navigation, as well as in computer graphics and animation.

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