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.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.