The accompanying table gives the number of reported deaths due to AIDS, during the t th year after 1995, for the period 1995 to 2006. (Source: Centers for Disease Control and Prevention, National Center for HIV, STD, and TB Prevention.)

However, the table does not tell the complete story because many deaths actually due to AIDS go unreported or are attributed to other causes. Let D(t) be the function that gives the cumulative number of AIDS deaths at time t. Then the data in the table can be thought of as rates of change of D(t) at various times, that is, as mortality rates. For instance, the table tells us that in 1997 (t = 2), AIDS deaths were increasing at the rate of 22,245 per year.

a. Assuming that D(t) is differentiable, explain why the total number of AIDSdeaths N during the period 1995–2006 is given by the integral

b. Estimate N by using the data in the table together with the trapezoidal rule to approximate the integral in part (a).

Transcribed Image Text:

Year 1995 1996 1997 1998 1999 2000 t 0 1 2 3 4 5 Reported AIDS Deaths 51,670 38,396 22,245 18,823 18,249 16,672 Year 2001 2002 2003 2004 2005 2006 t 6 7 8 9 10 11 Reported AIDS Deaths 15,603 16,948 16,690 16,395 16,268 14,016