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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

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.

To determine

To estimate:

The volume of the liver by using midpoint rule

Explanation

1) Concept:

i. If the cross section is a disc and the radius of the disc is in terms of x  or y then area A=π radius2

ii. The volume of the solid revolution about the x-axis is

V= abA(x)dx

iii. Use midpoint rule to find the volume of the liver

Midpoint Rule:abf(x)dx=Mni=1nfxi-x=x[fxi-+ .+ fxn-]

where  x=b - an, n is the number of subintervals and  xi-=12xi-1+xi is midpoint of [xi-1, xi]

2) Given:

Cross sectional areas in square centimetres are: 0, 18, 58, 79, 94, 106, 117, 128, 63, 39 and 0 and liver is 15 cm long. Also, cross sections are spaced 1.5 cm apart.

3) Calculation:

CAT scan of a human liver shows cross-sections spaced 1.5 cm apart, and the liver is 15 cm long

So, there are 11given points.

x 0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0
y 0 18 58 79 94 106 117 128 63 39 0

It is not possible to find y value for midpoint of (0, 1.5) therefore, consider first subinterval as (0, 3

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