How to Do Reimann Sums
One of the fundamental concepts in Calculus is Riemann sums, which can often confuse students. Riemann sums use simple geometry concepts such as the area of a rectangle to approximate the area of a curve! Once learned, Riemann sums can help you save time in finding the area under a curve. This blog will explain what they are, how to do them, and how to tackle different types of Riemann Sums.
What Are Riemann Sums?
In short, Riemann sums are an estimation of what the area under a line/curve would be. In order to do this, you divide everything below the curve into rectangles. By finding the area of each small rectangle and adding them together, you can estimate the total area under the curve.
The formula for this is Σ f(xi) * Δx.
Here, f(xi) is the y-value of the function at a specific point (x) for each rectangle. While the Δx is the width of each rectangle.
Although this formula seems daunting, once you learn how to use it, it will make sense.
How to Do Riemann Sums
1. Identify the number of subintervals
You are usually given this. It tells you how many rectangles to use to divide the whole interval.
2. Find the width of each rectangle
To find the width, divide the entire interval by the number of subintervals
3. Find the length of each rectangle
Each rectangle has a different length. Therefore, you need to use the point on the curve. You will learn how to determine which point to use in the next section.
4. Find the area of each rectangle
To find the area of each rectangle you must multiply the length by the width
5. Add each rectangle's area
To find the total approximation of the area under the curve, you must add the area of each rectangle.
Difference Between Left and Right Reimann Sums
Two of the main types of Riemann sums are left and right Riemann sums. Both are still approximations, however, they will result in different values. In left Riemann Sums, to find the length of the rectangle you would use the leftmost value on the curve as the height.
While in right Riemann Sums, you would use the rightmost value on the curve as the height of the rectangle.
Because each Riemann Sum (left and right) uses different height values, the approximation for the area under the curve would be different.
For example, in some cases using a right Riemann sum would give you an under-approximation while using a left Riemann sum would give you an over-approximation, and vice-versa.
Final Thoughts on Reimann Sums
Hopefully, this helped you understand Riemann sums and clear up questions you had about this concept. If you need more help on this or any other topic, go to UPchieve and request a session with a free tutor.