Midpoint Rule Confusion

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For example, if n = 4, you would have four rectangles with bases of 0.25, 0.5, 0.75, and 1.0In summary, the conversation discusses the use of the midpoint rule in approximating the length of a curve. It is clarified that the midpoint rule is different from Simpson's rule, which involves approximating the curve with parabolas. The midpoint rule involves calculating the function value at the midpoint of each subinterval and adding the lengths of the line segments connecting those points.
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Homework Statement


I'm working on a practice test and got stuck on the question: Approximate the length of the curve y=√x, 0≤x≤1 by the midpoint rule with n=4 and ∆x=0.25.


Homework Equations


Is the midpoint rule the same thing as Simpson's rule? I don't remember going over the midpoint rule in class and I can't find it in the book. If it isn't the same as Simpson's rule, what is it?
 
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I would guess that in finding the length of a curve by the midpoint rule, you calculate the function value at the midpoint of each subinterval, and then calculate the lengths of those line segments that join those points. For the approximate arc length, just add those lengths together.
 
  • #3
No, the midpoint rule is not the same as Simpson's rule. With Simpson's rule, you approximate the curve by a series of parabolas through each set of 3 points.

As Mark44 says, the midpoint rule uses the midpoint of each interval to compute the height of a rectangle on that interval.
 

1. What is the Midpoint Rule Confusion?

The Midpoint Rule Confusion is a common misunderstanding of the Midpoint Rule, which is used in numerical integration to approximate the area under a curve. It occurs when the midpoint is incorrectly chosen for a subinterval, leading to a significantly different result than the correct answer.

2. How does the Midpoint Rule work?

The Midpoint Rule divides the interval into subintervals and approximates the area under the curve by using the midpoint of each subinterval as the height of a rectangle. The sum of these rectangles is then calculated to estimate the total area under the curve.

3. What causes Midpoint Rule Confusion?

Midpoint Rule Confusion can be caused by several factors, such as using an incorrect formula, choosing the wrong number of subintervals, or not properly calculating the midpoint for each subinterval. It can also occur when the function being integrated is not continuous or when the interval is not evenly divided.

4. How can Midpoint Rule Confusion be avoided?

To avoid Midpoint Rule Confusion, it is important to carefully follow the steps of the Midpoint Rule and double-check all calculations. It is also helpful to use a graphing calculator or software to visualize the approximation and compare it to the actual area under the curve.

5. What are the implications of Midpoint Rule Confusion?

If Midpoint Rule Confusion is not caught and corrected, it can lead to significant errors in the estimated area under the curve. This can have consequences in various fields, such as engineering, physics, and economics, where numerical integration is commonly used to solve problems and make predictions.

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