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Riemann Sum Examples

Riemann Sum Calculator  

Find the left or right Riemann sum with a complete procedure of computation. The Riemann sum calculator allows the input of the subintervals and aids the inputting of the functions with a built-in keyboard.

How to use this Calculator?

The Riemann sum calculator requires the following steps to be completed.

  1. Enter the function and inbound limits.
  2. Select the side of the sum and variable.
  3. Input the subinterval and click Calculate.

What is the Riemann Sum?

A Riemann sum is a method used to approximate the definite integral of a function over a certain interval. 

To understand the idea behind the Riemann Sum, think of a curve on a graph. If we want to find the area under the curve over a certain interval, we can approximate it using rectangles. 

The idea is simple: split the interval into smaller subintervals, and for each of them, construct a rectangle. The sum of the areas of these rectangles gives an approximation of the area under the curve.

Essentially, you're dividing the region under a curve into smaller rectangles (or sometimes other shapes) and adding up the areas of these shapes. The more rectangles you use, and the narrower they become, the closer the approximation becomes to the actual value of the integral.

Why do we need it?

The Riemann Sum serves as the foundational idea for integration. The integral symbol ∫ actually represents an elongated "S," symbolizing a "sum." When we integrate, we are essentially taking the limit of the Riemann Sum as n approaches infinity — i.e., as the rectangles get infinitely narrow and numerous. This concept is essential in calculus, especially in dealing with the Fundamental Theorem of Calculus.

Formula for the Riemann sum:

If the interval [a,b] is divided into n equal subintervals of width Δx, and xi*is a point in the ith subinterval, then the Riemann sum is:

 Sn = i=1nf(xi*)Δx


  • Δx = b−a / n
  • xi* is any point in the ith subinterval.

Types of the Riemann sum:

There are three main types of Riemann sums:

  • Left Riemann Sum: Choose xi* as the left endpoint of each subinterval.
  • Right Riemann Sum: Choose xi* as the right endpoint for each subinterval.
  • Midpoint Riemann Sum: Choose xi* as the midpoint of each subinterval.

How to find the Riemann sum?

To compute a Riemann Sum, begin by selecting an interval [a,b] over which you want to approximate the area under a function f(x). Divide this interval into n equal subintervals, each with a width of Δx = b−a / n. 

Next, for each subinterval, pick a representative sample point xi*. The choice of this sample point determines the type of Riemann Sum: if you pick the left endpoint, it's a Left Riemann Sum; the right endpoint, a Right Riemann Sum; and the midpoint results in a Midpoint Riemann Sum. 

Evaluate the function at each chosen sample point to determine the height of the corresponding rectangle. The area of each rectangle is then f(xi*) × Δx.

Sum up the areas of all these rectangles to obtain the Riemann Sum, which provides an approximation of the area under the curve of the function over the chosen interval. The accuracy of this approximation typically improves as the number of subintervals n increases.

Solved Example:

Consider f(x) = x2 on the interval [0, 2] with n = 4.

Calculate Δx: This represents the width of each subinterval.

Δx = b−a / n

    = 2−0 / 4


Determine the Sample Points: For the Left Riemann Sum, we choose the leftmost points of each subinterval.

x1*= 0, x2*= 0.5, x3*= 1, x4*= 1.5

Evaluate the Function: We get the height of each rectangle by evaluating the function at the sample points.

f(0) = 0, f(0.5) = 0.25, f(1) =1, f(1.5) = 2.25

Compute the Area of Each Rectangle: For each rectangle, its area is the height (given by f(xi*)) multiplied by the width (Δx). Add up the areas of all rectangles.

Sn = 0(0.5) + 0.25(0.5) + 1(0.5) + 2.25(0.5) = 1.75

Thus, the Left Riemann Sum approximation for this interval and function with 4 subintervals is 1.75. As we increase n, our approximation will get closer to the actual area under the curve.

Real-Life Applications:

Physics: Calculating work done by a variable force, or the center of mass of objects with varying densities.

Engineering: Evaluating stresses and strains in materials with non-uniform density.

Economics: Finding total cost from a marginal cost curve or total profit from a marginal profit curve.

Environmental Science: Computing the total amount of materials required over time given a rate of use or the total pollutant produced given a rate of emission.

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