Midpoint Riemann sum approximation

In summary, a Midpoint Riemann Sum Approximation is a method used to estimate the area under a curve by dividing it into smaller rectangles and finding the sum of their areas, using the midpoint of each rectangle as its height. The formula for this approximation is Δx * [f(a+Δx/2) + f(a+3Δx/2) + f(a+5Δx/2) + ... + f(b-Δx/2)], where Δx is the width of each rectangle, a is the starting point, and b is the ending point. The purpose of using this method is to estimate the area under a curve when the function is not easily integrable, and it provides
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Leo Liu
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Homework Statement
This equation is found on PAGE 7, https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-3-the-definite-integral-and-its-applications/exam-3/materials-for-exam-3/MIT18_01SCF10_exam3sol.pdf
Relevant Equations
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1612402856532.png

Can someone please explain why the formula for midpoint approximation looks like the equation above instead of something like
$$M_n=(f(\frac{x_0+x_1}2)+f(\frac{x_1+x_2}2)+\cdots+f(\frac{x_{n-1}+x_n}2))\frac{b-a}n$$?
Thanks in advance!
 
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  • #2
In order to have enough partitions for the Simpson's Rule approximation, they have twice as many partitions as they are using for the midpoint approximation. So each of the midpoints that you are calculating by using an average is actually an exact partition point in their Simpson's Rule partitioning.
 
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1. What is a Midpoint Riemann sum approximation?

A Midpoint Riemann sum approximation is a method used in calculus to estimate the area under a curve. It involves dividing the area into smaller rectangles and finding the sum of their areas. The midpoint of each rectangle is used to calculate the height of the rectangle.

2. How is a Midpoint Riemann sum approximation calculated?

To calculate a Midpoint Riemann sum approximation, the interval of the curve is divided into smaller subintervals. The width of each subinterval is determined by dividing the total width of the interval by the number of subintervals. Then, the midpoint of each subinterval is found and used to calculate the height of the rectangle. The sum of the areas of all the rectangles gives an approximation of the area under the curve.

3. What is the purpose of using a Midpoint Riemann sum approximation?

The purpose of using a Midpoint Riemann sum approximation is to estimate the area under a curve when it is not possible to find the exact value using other methods. It is also helpful in visualizing and understanding the concept of integration.

4. What are the limitations of a Midpoint Riemann sum approximation?

One limitation of a Midpoint Riemann sum approximation is that it can only provide an estimate of the area under a curve, and the accuracy of the estimate depends on the number of subintervals used. It also assumes that the curve is continuous, which may not always be the case.

5. How does a Midpoint Riemann sum approximation compare to other methods of approximating area under a curve?

A Midpoint Riemann sum approximation is one of several methods used to approximate the area under a curve. Other methods, such as the Left Riemann sum and Right Riemann sum, use different points on the subintervals to calculate the height of the rectangles. The accuracy of the approximation may vary between methods, but they all follow the same principle of dividing the area into smaller rectangles.

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