Question

The 2010 census in a particular area gives us an age distribution that is approximately given (in millions) by the function f(x)=40.5+2.1x−0.732x² where x varies from 0 to 9 decades. The population of a given age group can be found by integrating this function over the interval for that age group. (a) Find the integral over the interval [0,9]. (Round to the nearest integer as needed.) What does this integral represent? A. The total number of people in this area aged 0 to 90 was about 272 million in 2010 . B. About 272 million people aged 0 to 90 have ever lived in this area from year 0 through 2010. C. The total number of poople in this area under 9 years old was about 272 in 2010. D. The number of people in this area increased by about 272 million people aged 0 to 90 in 2010.

Answer 1

The integral over the interval [0,9] of the function f(x)=40.5+2.1x−0.732x² represents the total number of people in this area aged 0 to 90 in 2010.

Step-by-step explanation:

The integral over the interval [0,9] of the function f(x)=40.5+2.1x−0.732x² can be found by integrating the function from 0 to 9. This can be done by finding the antiderivative of the function and evaluating it at the upper and lower limits of integration. The antiderivative of the function is F(x) = 40.5x + 1.05x² - 0.244x³. Evaluating this antiderivative at the limits of integration, we get F(9) - F(0) = 3645 - 0 = 3645 million.

Therefore, the integral over the interval [0,9] is approximately 3645 million. This integral represents the total number of people in this area aged 0 to 90 in 2010, so the correct answer is A. The total number of people in this area aged 0 to 90 was about 3645 million in 2010.