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-sections spaced 1.5 cm apart. The liver is 15 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 Rule to estimate the volume of the liver.
Applications of Integration
Alright, so I've taken the liberty of doing two things here, I've written down all of the values that we have for the cross sectional areas. We're here on the left side. And I have also written out there mid points here in the second column. Um and remember that to find the midpoint, we just add we just take the average of these two values. zero was 18 over to equals nine and 18 58/2 years, 30 years and we did that for all of these. Um it's rather time consuming, but this is going to be the result. Um After that, what midpoint rule tells us Is that in order to approximately the volume here we're going to need to uh take 1.5 which is the distance between the cross sections. And we multiply 1.5 x nine, Which is one minute point, then add to that 1.5 times 30 years. Then add to that 1.5 times 68.5 et cetera. And we can also uh do something somewhat simpler, which is first we add all of these midpoint values and then we multiply by 1.5 we just distribute us uh The 1.5. So um uh if you add up all these mid points you can do this by hand. I wouldn't recommend it, I just do a calculator but if you add all these up you are going to get 900 Which is pretty useful because that means that multiplying by 1.5 is very easy. 1.5 times 900 is just going to be 1350. So we're going to get uh, 1350. Now, of course, we need units for this. Can't just save 13.50. So we recall the units that we had, um, the areas were measured in square centimeters and then 1.5 was the distance between the cross sections in centimeters. So our units are just going to be cubic centimeters because we have spare centimeters multiplied by regular centimeters. So we have 13 50. Keep accelerators. That is going to be our approximation for the volume. Hey,