Setting up double integral for polar coordinates and integrating

In summary, The conversation is about setting up a double integral in polar coordinates to find the total population of a city. There is a discrepancy in the upper limit of integration for θ, with one person suggesting it should be from pi to pi/2 and another suggesting pi/2 to 3pi/2. The correct upper limit is determined to be pi/2 to 3pi/2, and this also affects the choices for part b, with the correct answer being either i or ii depending on the population density decreasing as distance from the shoreline increases. Part c involves using the answer from part b as the integrand for the double integral in part a.
  • #1
bradboynes
7
0
Link:

http://imageshack.us/photo/my-images/39/18463212.jpg/

This is a very long problem so I drew it to make things simpler.

Part a) tells me to set up a double integral in polar coordinates giving the total population of the city.

I have the following:
2π...4
∫...∫ δ(r, θ) r dr dθ. Is this correct?
π/2...1

For part b) I was plugging in numbers, but I feel it can be both i) and ii). Can someone help me figure this out?

For part c) all I would need to do is use what I find in part b) as the integrand for the double integral in part a).

Thank you!
 
Physics news on Phys.org
  • #2
If anyone can help me, I would be greatly appreciated.
 
  • #3
The upper limit of integration for θ is wrong.
 
  • #4
How is it wrong? if you look at it from the side, it should be from pi to pi/2. along the x-axis it's 0 degrees.
 
  • #5
Shouldn't it be from pi/2 to 3pi/2?
 
  • #6
It appears that Jaynte is correct.

In which case you should be able to choose between i & ii for pat (b).
 
  • #7
Yes: The population density, δ, decreases the farther you live from the shoreline.
 

1. What are the steps for setting up a double integral in polar coordinates?

The first step is to determine the limits of integration for both the angle, θ, and the radius, r. This will depend on the specific region or shape being integrated over. Next, rewrite the function being integrated in terms of r and θ. Then, multiply the function by r, which accounts for the extra "width" in polar coordinates. Finally, integrate the function with respect to both r and θ within their respective limits.

2. How do you convert a Cartesian function to polar coordinates?

The conversion from Cartesian coordinates (x, y) to polar coordinates (r, θ) follows the equations:
r = √(x² + y²) and θ = tan⁻¹(y/x).
This means that for a function f(x,y), it can be rewritten as f(r cos θ, r sin θ) in polar coordinates.

3. What is the purpose of multiplying the function by r in a double integral for polar coordinates?

In polar coordinates, the "width" of each infinitesimal element is not constant, as it is in Cartesian coordinates. The radius, r, changes as the angle, θ, changes. Multiplying the function by r accounts for this change in "width" and ensures that the integral is calculated correctly.

4. How do you determine the limits of integration for a double integral in polar coordinates?

The limits of integration for θ will typically range from 0 to 2π, as this covers the full circle. For r, the limits can be determined by looking at the shape or region being integrated over. The inner limit will be the smallest radius of the shape, while the outer limit will be the largest radius. If the shape or region is not centered at the origin, adjustments may need to be made to the limits.

5. Are there any special considerations when setting up a double integral in polar coordinates?

Yes, it is important to remember to include the appropriate Jacobian factor when converting the function from Cartesian to polar coordinates. The Jacobian is equal to r, so the function being integrated must be multiplied by r in order to account for this. Additionally, it is important to consider symmetry when determining the limits of integration, as this can often simplify the process.

Similar threads

Integration of acceleration in polar coordinates
    • Calculus and Beyond Homework Help
Replies
24
Views
1K
Change of Variables: Integrals w/Polar Coordinates
    • Calculus and Beyond Homework Help
Replies
2
Views
858
Double Integral Problem: Incorrect Jacobian Calculation for Polar Coordinates
    • Calculus and Beyond Homework Help
Replies
4
Views
901
Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
269
Double Integral in polar coordinates
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
Stokes' Theorem 'corollary' integral in cylindrical polar coordinates
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Problem with a double integral
    • Calculus and Beyond Homework Help
Replies
4
Views
868
Finding population density with double integrals
    • Calculus and Beyond Homework Help
Replies
11
Views
3K
Setting the limits of an integral
    • Calculus and Beyond Homework Help
Replies
3
Views
950
Converting a Cartesian Integral to a Polar Integral
    • Calculus and Beyond Homework Help
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top