The population density of a certain city is given by the function
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by

B)
C)
D)
E)

Question

The population density of a certain city is given by the function ​  ​
Where the origin (0, 0) gives the location of the government center. Find the population inside the rectangular area described by
​  ​
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

To find the population inside the rectangular area, we need to integrate the given function over the rectangular region.

The limits of integration for x are 0 to 1, and for y are 0 to 2.

∫[0,1] ∫[0,2] (1 + x + 2y) e^(-x-y) dy dx

= ∫[0,1] (1 + x + 4 - 4e^(-x)) e^(-x) dx

= [1 - 3e^(-x) + x - xe^(-x) + 4e^(-x)] [0,1]

= 11/2 - 2e^(-1)

≈ 2.27

Therefore, the population inside the rectangular area is approximately 2.27 million.

The closest answer choice is C) 2.3 million.

The Population Density of a Certain City Is Given by the Function