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.

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.

Transcript

00:01 So we want to estimate the area of the under the curve of f of t equals negative t times t minus 21 times t plus one in between t equals zero and t equals 12.

00:12 And we want to use six sub intervals and their midpoints to estimate this area.

00:18 So i find what's easiest to divide this out into six sub intervals is to give a visual.

00:25 So here we have t equals zero and two equals 12.

00:28 Now to divide this evenly into six pieces, we would need each piece to span a distance of t equals 2.

00:38 So here we have t equals 0, t equals 2, 4, 6, 8, and 10, and 12.

00:47 And then we also want to find their midpoints.

00:50 So to give you a better visual of that, we can draw rectangles between in, to represent each interval.

00:58 So there go.

01:02 And then their midpoints, the midpoint of this one would be one, because t equals 1 is halfway between t equals 0 and t equals 2.

01:11 So this midpoint would be 3, this 5, 7, 9, and 11.

01:25 So now that we know how to divide this into 6 submembrals and their midpoints, we will use the formula, the sum of, going 0 to 12, delta t, which is the change in t, and our function f of t sub i, or t sub i, is equal to our midpoint values.

01:57 So we know that our delta t, our change in t, is equal to 2.

02:02 So then we can write that into our formula.

02:06 0, 12, 2 times f of t sub i.

02:15 So we know we have six intervals, and we know the midpoint of each of those intervals, t sub i, being 1, 3, 5, 7, 9, and 11.

02:28 But we don't want to just want to multiply it by that number.

02:30 We want to know what the value of that function is at each of those midpoint values...

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