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How to do a Riemann Sum?

2 min readβ€’december 17, 2021

AP Calculus AB/BC ♾️

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Riemann Sums

A Riemann Sum estimates the area under a curve using rectangles. While this technique is not exact, it is an important tool that you can use if you are unable to differentiate or integrate an equation.Β Need more help, check out this other study guide for Riemann Sum explanation and practice!
So imagine you are given this equation: f(x) = x^2. Your interval is [0,5] and n = 5. For the purposes of simplicity, I am going to demonstrate a Left Riemann Sum.

1. Understand the information that the question gives you.

Equation: f(x) = x^2 Interval: [a,b] and in the case of this problem [1,5] - these are the parameters for the section under the curve that you are estimating N: n=4 - this number determines how many rectangles you are splitting the section under the curve into. (Note: More rectangles = more precision, but also means more work)

2. Find Ξ”x: the width of each of your rectangles.


3. Find the height of each rectangle: where the top left (for a Left Riemann Sum) corner of each rectangles meets the curve.

(**If you were to take a Right Riemann Sum you would use the top right corner of each rectangle and if you were to use a Midpoint Riemann Sum the height would be where the middle of each rectangle hit the curve)
Our heights for this specific problem would be 1, 4, 9, and 16

4. Find the area of each rectangle, and add them together.

(1)(1) + (1)(4) + (1)(9) + (1)(16) = 30Β 
A =Β Ξ”x( f(x1) + f(x2) + f(x3)...)
^In simplest terms, this equation will help you solve any Riemann Sum. Note that all the steps are the same for Right Riemann Sums except for #3. Just remember to use the top left corner of your rectangles for each Left Riemann Sum and the top right corner for each Right Riemann Sum.Β 

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