Riemann Sums

A Riemann sum is an approximation of an integral of a function (area under the curve) by the sum of a series. The series is generated by constructing rectangles where one corner touches the curve. Riemann sums can be used to approximate functions that cannot be integrated analytically. However, one of the more interesting applications of Riemann sums is the insight they can give on the convergence of infinite series which shall be discussed in the next page. What are Riemann sums exactly? we shall consider two types: left and right Reimann sums. The figures below show just one Reimann rectangle for increasing and decreasing f(x). Press the buttons to see all cases.

Left Riemann rectangle

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The area of the rectangle is simply the width time the height, where the height is f(x). The width and position of the rectangle can be changed. For decreasing f(x) the area of the rectangle is greater than the area under the curve i.e. the value of the integral. For an increasing f(x) the area of the rectangle is less than the area under the curve i.e the value of the integral.

Right Riemann rectangle

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For decreasing f(x) the area of a right Riemann rectangle is less than the area under the curve i.e. the value of the integral. For an increasing f(x) the area of the rectangle is greater than the area under the curve i.e the value of the integral. The left and right Riemann rectangles can be compared on this page.

Approximating Integrals

A large number of rectangles can be used to approximate an integral (area under the curve), this is shown below:

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The Riemann approximation for an integral can only be calculated using a computer program. The accuracy increases as the width of the strips decreases. The Riemann rectangles links a discrete series with a continous functions, and in the next we shall see how concept is exploited to determine if a series converges or diverges.