Finding population density with double integrals

[Mohamed Abdul]

Homework Statement

A city surrounds a bay as shown in Figure 1. The population density of the city (in thousands of people per square km) is f(r, θ), where r and θ are polar coordinates and distances are in km.

(a) Set up an iterated integral in polar coordinates to find the total population of the city.

(b) The population density decreases the farther you live from the shoreline of the bay; it also decreases the farther you live from the ocean. Which of the following functions best describes this situation?
i. f(r, θ) = (4 − r)(2 + sin θ)
ii. f(r, θ) = (4 − r)(2 − sin θ)
iii. f(r, θ) = (4 + r)(2 − cos θ)
Find the population using your answers to parts (a) and (b) above.

Here is the diagram the question refers to:

Homework Equations

I'm assuming population = double integral of population density.

The Attempt at a Solution

For the first part, I believe I should make a double integral with my bounds 1<r<4 and 0<theta<pi. In the integrand I'll simply put in f(r,θ). However, I'm not sure on the bounds for r. Do I use the lower and upper radius, or am I supposed to be finding the difference between the two circles depicted?

For the second part, I believe that iii would be correct since further distance would be marked with an increase in the r variable. I'm not too sure on this however

Also, I believe that the population can be calculated with the bounds and integral set up for a, but again, I'm unsure about my bounds for r.

Any help would be greatly appreciated, thank you.

 

[Mohamed Abdul]

I'm sorry, I forgot to add the relevant diagram. I've updated the OP, so if anyone could help me out I'd be very thankful.

 

[kuruman]

Your belief that you should set up a double integral is correct. Do you have an idea about how to do this? Hint: The number of people in an area element dA is dN = f(r,θ)dA. All you have to do is add all such area elements over the area where people live, i.e. where f(r,θ) is non-zero. That should guide your thinking about the bounds.

 

[Mohamed Abdul]

Your belief that you should set up a double integral is correct. Do you have an idea about how to do this? Hint: The number of people in an area element dA is dN = f(r,θ)dA. All you have to do is add all such area elements over the area where people live, i.e. where f(r,θ) is non-zero. That should guide your thinking about the bounds.

This is what I have so far, am I on the right track?

 

[kuruman]

I can't see what you have so far.

 

[Mohamed Abdul]

I can't see what you have so far.

Sorry, I updated my post

 

[kuruman]

You have correctly written the element dA = r dr dθ. Please read my post #3 again. For part (a) you are supposed to add elements dN, but you are actually adding elements dA.

 

[Mohamed Abdul]

You have correctly written the element dA = r dr dθ. Please read my post #3 again. For part (a) you are supposed to add elements dN, but you are actually adding elements dA.

For part a then, would I just put the f(r,theta)rdrdtheta stuff in there. But then I'm not sure what I would put for my final integral, since I'm not sure which of the integrals would make sense in the context of the problem.

 

[kuruman]

For part a then, would I just put the f(r,theta)rdrdtheta stuff in there.

That is correct.

I'm not sure which of the integrals would make sense in the context of the problem.

Figure out which of the three given distributions mathematically describes that
"The population density decreases the farther you live from the shoreline of the bay; it also decreases the farther you live from the ocean."

 

[Mohamed Abdul]

That is correct.

Figure out which of the three given distributions mathematically describes that
"The population density decreases the farther you live from the shoreline of the bay; it also decreases the farther you live from the ocean."

That would fit the second one: f(r, θ) = (4 − r)(2 − sin θ), since both parts of this function decrease with an increase with their respective variables. Or am I seeing this the wrong way?

 

[kuruman]

You are seeing it the right way. Now that you know the distribution all that remains is finding the population.

 

[Mohamed Abdul]

You are seeing it the right way. Now that you know the distribution all that remains is finding the population.

I'll get on that then, thank you very much