This is a word problem that deals with using rectangles to find the area under a curve, please help I am very lost!

Question

In someone infected 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 × 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.

Answer 1

You are being asked to approximate the exact area under the curve, which can be calculated with the definite integral ∫₀¹² f(t)dt, by adding the area of six rectangles. This approximation is known as a Riemann sum. The parameters that can change with a Riemann sum are the number of rectangles used, n, (in this case 6), and where we evaluate the function on each subinterval to get the heights of the rectangles (in this case, the midpoint).

The entire interval is t = 0 to t = 12, so divided equally into six subintervals gives us subintervals of width 2, that are t ∈ [0 , 2] , [2 , 4] , ... , [10 , 12]. The six midpoints of these subints are t = 1, 3, 5, 7, 9, and 11.

The area of the 1st rectangle is width·height = 2·f(1) = 2·40 = 80. The next area is 2·f(3), etc.

Since all the rectangles have width 2, the area approximation is 2 · [f(1) + f(3) + f(5) + f(7) + f(9) + f(11)].