How to Calculate Riemann Sums

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A Riemann sum is an approximation of the area under a mathematical curve between two X values. This area is approximated using a series of rectangles that have a width of delta X, which is chosen, and a height that is derived from the function in question, f(X). The smaller delta X is, the more accurate the approximation will be. The height can be taken from the value of the f(X) either on the right, middle or left of the rectangle. You can learn how to calculate a left-hand Riemann sum.

    Find the value of f(X) at the first X value. As an example, take the function f(X) = X^2, and we are approximating the area under the curve between 1 and 3 with a delta X of 1; 1 is the first X value in this case, so f(1) = 1^2 = 1.

    Multiply the height, as found in the previous step, by delta X. This will give you the area of the first rectangle. For the example, 1 x 1 = 1.

    Add delta X to the first X value. This will give you the X value at the left side of the second rectangle. For the example, 1 + 1 = 2.

    Repeat the above steps for the second rectangle. Continuing the example, f(2) = 2^2 = 4; 4 x 1 = 4. This is the area of the second rectangle in the example. Continue in this way until you have reached the final X value. For the example, there are only two rectangles because 2 +1 = 3, which is the end of the range being measured.

    Add the area of all the rectangles. This is the Riemann sum. Finishing the example, 1 + 4 = 5.


    • You may find drawing the function and rectangles to be helpful, but this is not necessary.


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Kaylee Finn began writing professionally for various websites in 2009, primarily contributing articles covering topics in business personal finance. She brings expertise in the areas of taxes, student loans and debt management to her writing. She received her Bachelor of Science in system dynamics from Worcester Polytechnic Institute.

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