The velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period.

In someone affected with measles, the virus level $ N $ (measured in number of infected cells per mL of blood plasma) reaches a peak density at about $ t = 12 $ days (when a rash appears) and then decreases fairly rapidly as a result of immune response. The area under the graph of $ N(t) $ from
$ t = 0 $ to $ t = 12 $ (as shown in the figure) is equal to the total amount of infection needed to develop symptoms (measured in density of infected cells x time). The function $ N $ has been modeled by the function $$ f(t) = -t(t - 21)(t + 1) $$

Use this model with six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms of measles.

The table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003.

(a) By using an argument similar to that in Example 4, estimate the number of people who died of SARS in Singapore between March 1 and May 24, 2003, using both left endpoints and right endpoints.

(b) How would you interpret the number of SARS deaths as an area under a curve?

Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.

$ f(x) = \dfrac{2x}{x^2 + 1}, \hspace{5mm} 1 \le x \le 3 $

Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.

$ f(x) = x^2 + \sqrt{1 + 2x}, \hspace{5mm} 4 \le x \le 7 $