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Problem 29 Medium Difficulty

A CAT scan produces equally spaced cross-sectional views
of a human organ that provide information about the organ
otherwise obtained only by surgery. Suppose that a CAT
scan of a human liver shows cross-scctions spaced 1.5 $\mathrm{cm}$
apart. The liver is 15 $\mathrm{cm}$ long and the cross-sectional areas,
in square centimeters, are $0,18,58,79,94,106,117,128,$
$63,39,$ and $0 .$ Use the Midpoint Rulc to estimate the volume of the liver.

Answer

1110 $\mathrm{cm}^{3}$

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Video Transcript

So one thing I can do to estimate volume is actually use a table and use real on sums so that rectangular approximation to find the volume. So ask me specific kind of table in order to do that. This table I was given really a length and then an area. And if you think about already a two dimensional area and then multiplying it by a length, that's what gives you that three dimensions. That makes it a volume. So that's like what stewed in a girl does when you have that two dimensional area and then you rotate it to make it into a volume. So in this case, let's use the midpoint rule for the table. So the midpoint rule says you want to look across Ah, whole interval. So I consider this an entire rectangle and you want to use the middle value the middle height given on this rectangle. Graphically, this is what you would see. You're 0 1.5 18 3 58 The midpoint rule is saying okay, I want to make a rectangle going from 0 to 3. But the height of that rectangle is only gonna be the middle of it, right, The midpoint of it, which is the 18. So a lot times I like to draw my rectangles on the table. It just makes less strong for me. So I want to think about what is the base of this rectangle and the base here is this Three units I go from 0 to 3 times the middle value would be 18. And that's what we saw when we had it sketch out on our graph. Right? The base of the rectangle was the 123 units we had here across. And then really, the height of that rectangle came from the 18 the middle value. And then I want to add this to the next directing. Also, I do it again. The basis still gonna be three units here times this middle value is 79 and I do it again to make another rectangle whose base is still three units. And I use the middle value, which is one of six plus another rectangle. And I just wanna do this all the way time through. So it's three and it's across the base. And then that middle output there is 1 28 and then lastly, rectangle that has three units across the base and that middle output is 39. So to figure out then the volume as this estimation, with the midpoint, I would calculate this all together, and that is 1110 and then use whatever units is given in this case, centimetres cubed is our units.

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