How to calculate the Area of a Curve using Riemann Sum is not exactly technical? The Riemann sum is a type of approximation technique that can be used to find the area of a function. However, its primary application is to determine a definite integral. Now in Calculus, the Integral of a function can be used to determine its area as long as the integral interval is definite. This particular application is one of the major and most accurate ways to determine the Area of a Function.
However, there are times when determining the area of a function using an integral may not be possible. This impossibility may be due to the difficulty of applying the theorem of integration for that function. When that is the case, The Riemann sum becomes one of the best alternatives to determine the function area. However, the answer that will be reached is usually an approximation of what the integral would have given if it could be solved.
The Riemann sum can be calculated in three major ways and that is by determining the Left Riemann Sum, Right Riemann Sum, and Mid-point Riemann sum.
This article will consider what the Riemann sum is and its calculations
What Is Riemann Sum?
The Riemann sum as an approximation simply considers how a graphed function region looks and divides the focus area (Area of A curve) into partitions. These partitions could be from anywhere to two or more. Each partition is then considered as a shape. The rectangle is the most popular shape of a partition used for partitioning a focus Area. Now each rectangle area (L x w) is calculated and joined together. Since all of the rectangles were actually formed from the curve of the function, their answer is then considered the entire curve’s area. Albeit, this improvised answer is only an approximation.
The shapes that the partitioning of an area of a curve may form could be Rectangle, Triangles, Parabolas, Trapezoids and more.
An example of what is shown in the graphs below
If the above is a graphed function with the curve determining the function area, then the Riemann sum approximation with shapes of the same curve will look something like this
Riemann Sum Formula
The Riemann Sum is approximated solution of an Integral that has definite sums.
Let the graph below represent a function F(x) and let the rectangles be the shapes partitioning the entire area. Also, let the starting point of the first triangle be the starting point of the area, and the endpoint of the last part of the area, the addition of all the triangle areas will be the Riemann approximated solution of the area.
Now, the formula for the area of a rectangle is length X Breadth (LXB). However, in this case, the B will be represented by
, while the L will be represented by Y= f(x*)
Therefore each area of the rectangles that form the entire area to be calculated is
Since the Riemann sum of all the rectangles is their addition, then the formula is
where n is the nth (last) triangle that completes the summation
is the general Riemann equation or formula
is a constant as all rectangle breath are going to be the same. Its formula for is,
where a is the lower bound of the definite sum and B is the upper bound.
However, since the Riemann sum can be computed from both the Left (left Riemann sum), Right (Right Riemann sum) or middle points (Mid-point Riemann sum), it, therefore, means that the points to be used for the calculations will usually vary.
However, since the Riemann sum can be computed from both the Left (left Riemann sum), Right (Right Riemann sum) or middle points (Mid-point Riemann sum), it, therefore, means that the points to be used for the calculations will usually vary. The Left Riemann formula is
The Right Riemann formula is
The Mid-point Riemann formula is
Now it is important to opt for the mid-point rule compared to the Left and right rule. This is because, in most cases, their values could be far off the actual solution. The number of rectangles influences how close the value of the Left or right rule will properly approximate the solution. The higher the number of triangles, the better the approximation for the two rules. For the Mid-point rule, this is not the case as the number of rectangles does not affect the solution.
How to Calculate Riemann Sum?
Step 1: Determine the Formula
Step 2: choose the relevant Riemann Sum out of the Left, Right or Midpoint variant
Now we will show with an example how the Riemann sum works. In this example, we will use a function that can be integrated so that we can get its solution and also use the Riemann sum to show that its approximated area is indeed similar to the integral.
Let the function be in the form of the graph with four rectangles determining the four partitions.
Then the Area of the integral is;
Using the basic integral formula of
Since there is a bound of [0,2]
Now we will use the Riemann Sum (Left, Right and Mid-point) to find an approximated answer
The area below the curve is under approximation
The Riemann sum formula is
Since the interval is from 0 to 2 and there are 4 rectangles, then
Since their 4 rectangles in between 0 and 2 and the width of each rectangle is 0.5, then all the points will be;
0, 0.5, 1, 1.5, 2
For the left Riemann sum, only the first 4 numbers from the Left will be used, which is 0, 0.5, 1, and 1.5,
For the Right Riemann sum, only the last 4 numbers will be used, which are 0.5, 1, 1.5, and 2
For the midpoint Riemann sum,
As such only the middle numbers in the generated number line will be the main numbers. i.e (0 + 0.5)/2, (0.5 + 1)/2, (1 + 1.5)/2 mad (1.5 + 2)/2 = 0.25, 0.75, 1.25, 1.75
Since all of the numbers for all of the Riemann sum have been determined they will be computed one after the other
The Left Riemann Sum
Clearly, this approximation is not close to the 4.67 solutions as the rectangle is not so many.
Now for the Right Riemann sum,
Since the numbers are 0.5, 1, 1.5, and 2
Again the approximated solution is not close to the actual solution.
Now the mid-point
Since 0.25, 0.75, 1.25, 1.75
Apparently, 4.625 is a good approximation of the solution 4.67
From the above, it is clear why it is advised for students to use the mid-point Riemann sum for approximation.
Calculating the Riemann sum is not very difficult and only requires a good understanding of the formulas work. Once a student has been able to determine the points for any of the Riemann sum methods of calculation, then almost half of the entire problem would have been handled.
This piece considered an example and outlined how people can use any of the three choices to get their approximated solutions.