# Midpoint rule causing confusion

1. Oct 15, 2014

### Lord Anoobis

1. The problem statement, all variables and given/known data
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 1.5cm apart. The liver is 15cm long and the cross-sectional areas in square centimetres, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39 and 0. Use the midpoint rule to estimate the volume of the liver.

2. Relevant equations

3. The attempt at a solution
I ended up with an answer of 1053cm3 by taking the midpoints of the following pairs of values, (0, 18), (18, 58), (58, 79), (79, 94), (94, 106), (106, 117), (117, 128), (128, 63), (63, 39), (39, 0), Resulting in

V = 1.5 x ( 9 + 38 + 68.5 + 86.5 + 100 + 111.5 + 122.5 + 95.5 + 51 + 19.5 )
= 1.5 x 702
= 1053cm3

The correct answer is 1110cm3. 1053cm3 seems to be the answer for either the left or right endpoint. I just know I've done something silly here but I can't spot where the error lies.

2. Oct 15, 2014

### Staff: Mentor

I've checked your work, and I don't see anything wrong, either.

3. Oct 15, 2014

### Lord Anoobis

Probably a printing error then. Thanks for the input.

4. Oct 15, 2014

### RUber

5. Oct 15, 2014

### Staff: Mentor

I agree with RUber's work.

6. Oct 17, 2014

### Lord Anoobis

At first, using a 3cm interval made no sense. However, once I made a plot of the whole works to see what was happening it became crystal clear!
With no function describing the process, only alternate points can be used since only those are known to lie on the curve with the available information, correct? I feel that while this is something of a trick question, it does carry an important lesson. One which will not be forgotten. Many thanks.

7. Oct 1, 2015

### kjfd

You can't make up your own midpoints, since you don't have a function to prove.
So, the solution is:
+((15-0)/5)(18+79+106+128+39)= 1110 cm^3