The population density

Question

The population density of a city r miles from the city center is P(r) people per square mile. Which expression represents the total number of people living within a circular region extending 5 miles from the city center?

Answer 1

Imagine the city divided into very many thin concentric circular strips.

A typical strip is some distance r from the center and has very small thickness dr.

It is bounded by two circles -- one with radius r and the other with radius r+dr.

The area of that strip is approximately 2πr×dr.

In reality it is a little larger, but we can approximate the area as 2πr×dr,

because the error goes to 0 as dr goes to 0.

The population in that strip is 2πr×dr×P(r).

The population in the whole city is the sum of all those circular strips,

with the radius of the strips ranging from 0 to 5.

Therefore the population of the city is the second integral.