Calc 2 Integration Word Problem. (3 parts) Help please

[hanjy]

The Problem below is 3 parts and I don't know how to start any of the parts. For part 1 do I need to plug in values for x or mA and just try to solve the integration from 0 to mA?

Imagine a country where everyone works for the government and the government has a finite amount of money to give out in annual salaries (total of I dollars). The entire population is P, but not all of these people work (e.g. children, elderly, disabled, etc.)

Let A be the average income of the citizens. The highest possible income will be mA for some constant m.

For each income level x, from 0 to mA, let f(x) be the fraction of the population that earns no more than x dollars per year.
^mA
1. Show that A = ∫ xf^′(x)dx
⁰
(We will assume that f(x) is differentiable)

Let D be the difference between the average income and largest income. i.e. D = mA − A
mA
2. show that D = ∫ f(x)dx
0

3. If the each income level has the same number of people, we will that there is a fair distribution of income. Show that in this case, f(x) is linear
i.e. f(x) = ax + b for some constants a and b
hint: show that f′(x) is a constant

 

[hanjy]

Any hints on how to start any of the parts is much appreciated because I have no idea how to approach this.

Imagine a country where everyone works for the government and the government has a finite amount of money to give out in annual salaries (total of I dollars). The entire population is P, but not all of these people work (e.g. children, elderly, disabled, etc.)

Let A be the average income of the citizens. The highest possible income will be mA for some constant m.

For each income level x, from 0 to mA, let f(x) be the fraction of the population that earns no more than x dollars per year.

mA
1. Show that A = ∫ xf′(x)dx (should say the integration from 0 to mA)
0
(We will assume that f(x) is differentiable)

Let D be the difference between the average income and largest income. i.e. D = mA − A

mA
2. show that D = ∫ f(x)dx (should say the integration from 0 to mA) (should read the integration from 0 to mA)
0

3. If the each income level has the same number of people, we will that there is a fair distribution of income. Show that in this case, f(x) is linear
i.e. f(x) = ax + b for some constants a and b
hint: show that f′(x) is a constant

 

[Deleted member 4993]

Any hints on how to start any of the parts is much appreciated because I have no idea how to approach this.

Imagine a country where everyone works for the government and the government has a finite amount of money to give out in annual salaries (total of I dollars). The entire population is P, but not all of these people work (e.g. children, elderly, disabled, etc.)

Let A be the average income of the citizens. The highest possible income will be mA for some constant m.

For each income level x, from 0 to mA, let f(x) be the fraction of the population that earns no more than x dollars per year.

mA
1. Show that A = ∫ xf′(x)dx (should say the integration from 0 to mA)
0
(We will assume that f(x) is differentiable)

Let D be the difference between the average income and largest income. i.e. D = mA − A

mA
2. show that D = ∫ f(x)dx (should say the integration from 0 to mA) (should read the integration from 0 to mA)
0

3. If the each income level has the same number of people, we will that there is a fair distribution of income. Show that in this case, f(x) is linear
i.e. f(x) = ax + b for some constants a and b
hint: show that f′(x) is a constant

What are your thoughts?

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[hanjy]

1. for part 1 I think I should use the formula E=[X] = ∫xfX(x)dx but I don't know how to set up the problem after that

2. For part 2 I think I use integration by parts and I got up to xf(x)-∫x'g(x)dx but not sure where to go form there.

[FONT=MathJax_Math][/FONT]where the integration is over the entire range of support of [FONT=MathJax_Math]X[/FONT]