## What is Riemann Sum?

Mathematicians have struggled for years with the problem of computing the area under a curve, or the region between a function and the x-axis. German mathematician Bernhard Riemann created the Riemann sum, also known as the Riemann approximation, which is the sum of vertical slices of the area in question, before calculus was established to compute those areas algebraically with integrals. However, because they have straight edges to make them simpler to compute, the slices are not precisely cut.

Rectangles are typically used in Riemann sums, however one type also uses trapezoids. When the function is positive, it passes through the tops of the rectangles or trapezoids; when it is negative, it passes through the bottoms of the rectangles or trapezoids. The x-axis is where the opposite sides of the rectangles or trapezoids are located.

The accuracy of a Riemann sum depends on the type of Riemann sum being used and how exactly the area will be calculated because the slices are not perfect. Because the distance between the straight-line approximation and the actual curve is lower for thinner slices, the accuracy of the sum increases as the slice width decreases.

## What is Riemann Sum Formula?

A is the area under the curve on the interval being evaluated, f(xi)f(xi) is the height of each rectangle (or the average of the two heights in the case of a trapezoid), and xx is the width of each rectangle or trapezoid. This formula is known as the Riemann sum.

The area of a rectangular slice in a Riemann sum is f(xi)xf(xi)x because the area of a rectangle is equal to the product of its base and height. The total area of the rectangles can be calculated by adding those areas.

The area of a trapezoidal slice is also f(xa)+f(xb)2xf(xa)+f(xb)2x since the area of a trapezoidal slice in a Riemann sum equals the product of its width and the average of its two heights (f(xa)f(xa) and f(xb)f(xb)). The total area of the trapezoids can be calculated by adding those areas.

The formula can be applied even if not all rectangles or trapezoids have the same width as long as the appropriate width is utilized for each shape.

## How to Use Formula of Riemann Sum?

You can solve the definite integrals as well as sample points by using online riemann sums calculator. The online tool follow the same formula. The formula by carrying out the actions listed below:

- Choose the number of slices to utilize and one of the four Riemann sum types (left, midway, right, or trapezoidal).
- To determine the breadth of each shape, divide the width of the interval by the number of slices (or find the width of each shape individually if not using a uniform width).
- Sketch the rectangles or trapezoids for that type of sum once you have drawn the function.
- To determine the heights, enter the x-values of the vertical lines into the function.
- Multiply the height by the width for each shape (or the average of the two heights for a trapezoid).
- Add the items from step 5 here.

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## Types of Riemann Sum:

### Â· **Left Riemann Sum**

Points on a function are used as the left vertices of rectangles that are perched on the x-axis in a left Riemann sum. In picture 4, there are four slices of identical width that represent the region under the curve from x = 0 to x = 2. Because x = 0 is the left endpoint of the interval being measured, a vertical line is drawn from the x-axis to the function at that point (in this instance, the vertical line at x = 0 has a height of 0, but that is not always the case). The dividing lines are drawn as vertical lines from the x-axis to the function at x = 0.5, x = 1, and x = 1.5.

### Â· **Center Riemann Sum**

Points on a function are used as the midpoints of the tops or bottoms of rectangles that are supported by the x-axis in a midpoint Riemann sum. In figure 5, the area under the curve is divided into 16 slices of identical width from x = -8 to x = 8. At the ends of the interval being measured and in between each of the 16 slices, vertical lines are drawn from the x-axis to the function. Rectangles are formed by drawing horizontal lines so that the middle of the top or bottom (depending on which side is off the x-axis) of each rectangle lies on the function.

### Â· **Right Riemann Sum**

Points on a function are used as the right vertices of rectangles that are perched on the x-axis in a right Riemann sum.

In picture 6, there are four equal-width slices representing the region under the curve from x = 0 to x = 2.

Between the four slices and the right endpoint of the interval being measured, vertical lines are drawn from the x-axis to the function. Rectangles with right vertices on the function are produced by drawing horizontal lines to the left from the aforementioned points on the function.

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## Riemann Sums Rules:

The Left Hand Rule, the Right Hand Rule, and the Midpoint Rule are three commonly used methods for calculating the height of these rectangles:

- According to the Left Hand Rule, the rectangle should be that height after evaluating the function at the subinterval’s left-hand endpoint. The Left Hand Rule’s height, f(2)=4, is used to create the rectangle with the designation “LHR” on the interval [2,3][2,3].

- According to the Right Hand Rule, the rectangle should be that height on each subinterval after evaluating the function at the right endpoint. The Right Hand Rule’s height, f(1)=3, is used to create the rectangle with the label “RHR” on the range [0, 1].

- According to the Halfway Rule, the rectangle should be that height on each subinterval after evaluating the function at its midpoint. The Midpoint Rule was used to calculate the height of the rectangle with the label “MPR,” which is drawn on the range [1,2][1,2] with the value f(1.5)=3.75.

Although we are not required to apply one of these three approaches, these are the three most popular rules for figuring out the heights of approximate rectangles. The height of the rectangle labelled “other” is chosen at random from among the values of the interval [3,4] and is drawn on the interval [3,4]. [3,4]. The selected xx-value is 3.54, which results in a height of f. (3.54).