In someone infected with measles, the virus level N...

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 $\times$ 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.

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 $\times$ 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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Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities, such as areas under a curve, over a specific interval. It is a fundamental concept in calculus used to compute the total change or quantity when a continuous function is integrated over an interval.

Riemann Sum

A Riemann sum is an approach to approximate the value of a definite integral by dividing the interval into subintervals and summing up the areas of rectangles whose heights are determined by the function's values at chosen sample points. This method forms the basis for numerical integration techniques.

Midpoint Rule

The midpoint rule is a numerical integration technique that estimates the area under a curve by evaluating the function at the middle point of each subinterval. By using the midpoint, this rule often provides a more accurate approximation of the integral compared to using an endpoint, especially when the function behaves relatively smoothly.

Subinterval Partitioning

Subinterval partitioning involves dividing the overall interval into a number of smaller, equal segments, which facilitates the application of numerical integration methods like Riemann sums. This division allows for a systematic approximation of integrals by enabling the evaluation of the function at specific points corresponding to each segment.

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