How Can the Midpoint Rule Estimate Liver Volume from CAT Scan Data?

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In summary, the midpoint rule is a numerical method used in calculus to approximate the area under a curve. It involves dividing the area into smaller rectangles and finding the sum of their areas. The formula for the midpoint rule is Δx * [f(x<sub>1</sub> + f(x<sub>2</sub> + ... + f(x<sub>n</sub>)] and its accuracy depends on the number of rectangles used. It is a simple and efficient method for approximating areas, but its accuracy may be affected by the shape of the curve.
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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, 17, 59, 77, 94, 106, 117, 127, 64, 39, and 0. Use the Midpoint Rule with n = 5 to estimate the volume V of the liver.


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FAQ: How Can the Midpoint Rule Estimate Liver Volume from CAT Scan Data?

What is the midpoint rule and how is it used?

The midpoint rule is a numerical method used in calculus to approximate the area under a curve. It involves dividing the area into smaller rectangles and finding the sum of their areas, which is a close approximation to the actual area under the curve.

What is the formula for the midpoint rule?

The formula for the midpoint rule is: Δx * [f(x1 + f(x2 + ... + f(xn)] where Δx is the width of each rectangle and n is the number of rectangles used.

How accurate is the midpoint rule?

The accuracy of the midpoint rule depends on the number of rectangles used. The more rectangles, the closer the approximation will be to the actual area under the curve.

What are the advantages of using the midpoint rule?

The midpoint rule is a simple and easy-to-use method for approximating areas under curves. It also provides a good balance between accuracy and computational complexity.

Can the midpoint rule be used for any type of curve?

The midpoint rule can be used for any curve as long as the function is continuous. However, for curves with sharp turns or irregular shapes, the accuracy may be lower and more rectangles may be needed for a better approximation.

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