Mastering Riemann Sums: How to Estimate Area Under a Curve with Ease

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In summary, the conversation is about using Riemann sums to estimate the area under the curve of f(x) = x^2 between x=2 and x=12. The speaker has been struggling to understand Riemann sums and is seeking help. The website provided in Section 2 calculates the upper Riemann sum for x^2 between 1 and 4, and the speaker notes that the lower Riemann sum can be found by subtracting the last term and adding an appropriate first term.
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maccaman
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ive been searching for ages on how to do Reimann sums, and none of it makes any sense to me compared to other forms of integration. My problem is i have to use Riemann Sums to estimate the area under the curve f(x) = x^2 between x=2 and x=12. Any help would be hugely appreciated, Thanks
 
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thanks heaps for the post, really helps me, cheers
 

1. What is a Riemann Sum?

A Riemann Sum is a method of approximating the area under a curve by dividing it into smaller rectangles and calculating their individual areas, then summing them together.

2. How are Riemann Sums used in calculus?

Riemann Sums are used in calculus to find the area under a curve when the function is not easily integrable. They are also used to define and calculate the definite integral of a function.

3. What is the difference between a left, right, and middle Riemann Sum?

A left Riemann Sum uses the left endpoints of each rectangle to calculate its area, a right Riemann Sum uses the right endpoints, and a middle Riemann Sum uses the midpoint of each rectangle. The choice of Riemann Sum method can affect the accuracy of the approximation.

4. How do you calculate a Riemann Sum?

To calculate a Riemann Sum, you first need to divide the interval into smaller subintervals of equal width. Then, choose a Riemann Sum method and calculate the area of each rectangle using the function values at the chosen endpoint or midpoint. Finally, add up all the individual rectangle areas to get the approximation of the area under the curve.

5. What is the significance of Riemann Sums in real-world applications?

Riemann Sums are used in many real-world applications, such as calculating the work done by a variable force, finding the distance traveled by a moving object with varying speed, and estimating the volume of irregularly shaped objects. They are also used in economics and finance to approximate the total profit or loss from varying prices.

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