Question

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, 59, 78, 93, 105, 118, 128, 63, 38, and 0. Use the Midpoint Rule with n = 5 to estimate the volume V of the liver.

Answer 1

Answer:

$V=1101cm^3$

Step-by-step explanation:

You are given this data:

$\left[\begin{array}{cccccccccccc}long&0&1.5&3&4.5&6&7.5&9&10.5&12&13.5&15&&\\Area&0&18&59&78&93&105&118&128&63&38&0\end{array}\right]$

First, calculate the x points by dividing the total length in 5:

$\Delta{x}=\frac{l_f-l_0}{5}= \frac{15-0}{5}= 3$

x=3,6,9,12,15

Now you calculate the half point of the x axis intervals you just calculated:

$x_h=1.5,4.5,7.5,10.5,13.5$

and find the function values of each of them (the Area for each cut):

A(1.5) = 18

A(4.5)=78

A(7.5)=105

A(10.5)=128

A(13.5)=38

Now you have formed the rectangles (see diagram below).

To calculate the volume, just use the next equation given by the midpoint rule:

$V=\Delta{x}\sum_1^5{h_{rectangle}}\\V=3\sum(18, 78, 105, 128, 38)\\V=3(367)\\V=1101cm^3$