Question

The density of people (number of people per mile) during the evening rush hour for the 5 mile stretch along a certain sidewalk in New York is given by f(x), where x is the distance in miles north of the subway station. Which of the following gives the number of people on this 5 mile stretch from the subway?

5
a. ∫ f(x). dx
0

x
b. ∫ f(x). dx
0

x
c. ∫ f(x). dx
5


5
d. ∫ f(x). dx
x

Answer 1

a.
`\int\limits^5_0 {f(x)} \, dx`

Explanation:

Since f(x) is the function for the populational density at a certain sidewalk for a 5 mile stretch, a definite integral of that function will yield the total number of people within the integration intervals. If we are interested in the number of people in the whole 5 mile stretch, we must integrate f(x) from x = 0 miles to x = 5 miles:

`\int\limits^5_0 {f(x)} \, dx`

Therefore, the answer is alternative a.