# Circle City, a typical metropolis, is densely populated near its center, and its population...

## Question:

Circle City, a typical metropolis, is densely populated near its center, and its population gradually thins out toward the city limits. In fact, its population density is 10,000(3 - r) {eq}people/mi^2 {/eq} at distance r miles from the center.

(a) Assuming that the population density at the city limits is zero, find the radius of the city.

(b) What is the total population of the city?

## Integrating Density

If we have a function that tells us the density or concentration of a quantity, integrating that function over a set of bounds tells us the total amount of that quantity that lies between those bounds.

a) Since the density will be zero at the city limits, we can set this density function equal to zero and solve for r to find the radius of the city.

{eq}10000(3-r) = 0\\ 3-r = 0\\ r = 3 {/eq}

The city has a radius of 3 miles.

b) As this is a density function that tells us the population at any radius from the city center, we can find the total population of the city by integrating this function over the distance from the center to the outer limits of the city, which is from 0 to 3 miles. As this is a linear function, we can find the exact population using antiderivatives.

{eq}\begin{align*} \int_0^3 10000(3-r) dr &= \int_0^3 (30000-10000r) dr\\ &= 30000r - 5000r^2 |_0^3\\ &= (30000(3) - 5000(3)^2) -(30000(0) - 5000(0)^2)\\ &= 45000 \end{align*} {/eq}

The total population of this city is 45,000.