Riemann sum

Four of the Riemann summation methods for approximating the area under curves. Right and left methods make the approximation using the right and left endpoints of each subinterval, respectively. Maximum and minimum methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively. The values of the sums converge as the subintervals halve from top-left to bottom-right.

In mathematics, the Riemann sum is a method of approximating the integral of a function over an interval. One representation of the integral, is the area under a curve in a graph. The Riemann sum is calculated by splitting the region into equal parts. In each part, the function is replaced by a shape. Shapes that are commonly used are rectangles, trapezoids, parabolas and cubics.The area of the similar shape can then be calculated. This method is often used when finding the closed form integral is difficult or impossible. The shapes used also influence how accurate the approximation is. More complex shapes give a higher accuracy, but this also means that calculating the approximation becomes more difficult.

Because the region filled by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing up the region more finely, using smaller and smaller shapes. As the shapes get smaller and smaller, the sum approaches the Riemann integral.

The Riemann sum may also be used to define the integration operation. The sum is named after a German mathematician who was called Bernhard Riemann.

Definition

[math]\displaystyle{ \text{Area} = \sum_{i=1}^{n} f(y_{i})(x_{i}-x_{i-1}) }[/math]

You divide the horizontal length under the part of the function you want to evaluate into "n" equal pieces. That is the n on top of the Σ (Greek letter sigma). The (x_i-x_i-1) represents the size of one horizontal segment that is created from dividing the whole by the "n". The f(y_i) is a y value in an "n" segment. Since the area of a rectangle is length × width, the multiplication of x_i and f(y_i) is the area of a rectangle for that part of the graph. The Σ means we add up all of these small rectangles to get an approximation of the area under the segment of a function.

Riemann Sum Media

• 

  Four of the methods for approximating the area under curves. Left and right methods make the approximation using the right and left endpoints of each subinterval, respectively.

• 

  Left Riemann sum of x ↦ x3 over [0, 2] using 4 subintervals

• 

  Right Riemann sum of x ↦ x3 over [0, 2] using 4 subintervals

• 

  Middle Riemann sum of x ↦ x3 over [0, 2] using 4 subintervals

• 

  Trapezoidal sum of x ↦ x3 over [0, 2] using 4 subintervals

Other websites

• A simulation showing the convergence of Riemann sums