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Problem 19 Easy Difficulty

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.

Answer

7840

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Video Transcript

Okay we want to find that area under this curve from T. Equals 02 T. Equals 12. And we're going to do it by making six uh intervals. And then use the mid points to find the high. Okay so first we want to find out how why the intervals are delta X. So that'll be 12 0 divided by six. Which means there to each. Hopefully you're not surprised by that. Okay so then Here's what it would look like. zero 2468 1012. Okay. I did pretty good that time usually run out. Okay and the reason I want to do that is because what we're interested in is the midpoint of those. So the mid points are 13579 and 12. Uh no 11. Sorry? Okay so now what we're gonna do is we're gonna plug all those mid points into this function and find out what its value is. So what we're finding is a rectangle really? Who's with this too? And its height is F. Of whatever you have here then we're going to just add up all those rectangles. Okay So f. of Lawn will be one times 20 times. Is that T. Plus we're not better write that down again here. T. G -21 G. Plus one. So it's more on time. What's T. -21? So one and yeah yeah -20 and two. So did I copy the problem? Right okay minus key. So minus one minus 20 and two which will make this one positive 40 and then F. Of three that will be negative three times negative 18 times four. So it looks yeah 1212 Times 18. So to 16 and then F of five will be minus five times negative 16 times six. So 30 times 16 there'll be 4 80 then F of seven will be -7 times -14 times eight. So 56 times 14 7 84 F. Of nine The -9 times -12 times 10. So 90 times 12 1080 And then if of 11 will be negative 11 times -10 times 12. So 11 time it's from 20 13 20. Yeah. Okay so that's the height of each rectangle. And then we're gonna multiply that times the width, height comes with which is the width of each interval is two. Let's be 84 30 to 9 60 calculator on that. 7 84 times two. Whoops. 784 Times two, 15 68 2160. 26 40. So when we add that up and we're going to find is our estimate of the midpoint or I mean our estimate of the area. So 80 Plus 4 32 plus 9 60 Plus 1568 plus 21 60 plus 26 40 78 40. Okay so I get 78 40 is the area under the curve. So that should be the answer to this problem