Riemann sum

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In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It may also be used to define the integration operation. The sums are named after the German mathematician Bernhard Riemann.

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[edit] Definition

Consider a function f: D → R, where D is a subset of the real numbers R, and let I = [a, b] be a closed interval contained in D. A finite set of points {x0, x1, x2, ... xn} such that a = x0 < x1 < x2 ... < xn = b creates a partition

P = {[x0, x1), [x1, x2), ... [xn-1, xn]}

of I.

If P is a partition with n elements of I, then the Riemann sum of f over I with the partition P is defined as

S = \sum_{i=1}^{n} f(y_i)(x_{i}-x_{i-1})

where xi-1 ≤ yi ≤ xi. The choice of yi in this interval is arbitrary. If yi = xi-1 for all i, then S is called a left Riemann sum. If yi = xi, then S is called a right Riemann sum. If yi = (xi+xi-1)/2, then S is called a middle Riemann sum. By averaging the left and right Riemann sum one obtains the so-called trapezoidal sum.

Suppose we have

S = \sum_{i=1}^{n} v_i(x_{i}-x_{i-1})

where vi is the supremum of f over [xi-1, xi]; then S is defined to be an upper Riemann sum. Similarly, if vi is the infimum of f over [xi−1, xi], then S is a lower Riemann sum.

Any Riemann sum on a given partition (that is, for any choice of yi between xi-1 and xi) is contained between the lower and the upper Riemann sums. A function is defined to be Riemann integrable if the lower and upper Riemann sums get ever closer as the partition gets finer and finer. This fact can also be used for numerical integration.

[edit] Methods

There are four common methods to compute a Riemann sum: left, right, middle, and trapezoidal. We will elaborate on them in the simple case when the partition is made up of intervals of equal size. Thus, divide the interval [a, b] into n subintervals, each of length Q = (b − a) / n. The points in the partition will then be

a, a + Q, a + 2Q, ..., a + (n−2)Q, a + (n−1)Q, b.

[edit] Left Riemann sum

A left Riemann sum of x3 over [0,2] using 4 subdivisions.

For the left Riemann sum, we will approximate the function by its value at the left-end point. This gives multiple rectangles with base Q and height f(a + iQ). Doing this for i = 0, 1, ..., n−1, and adding up the resulting areas gives us

Q\left[f(a) + f(a + Q) + f(a + 2Q)+\cdots+f(b - Q)\right].\,

The left-hand Riemann sum will be an overestimation if f is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing.

[edit] Right Riemann sum

A Right Riemann sum of x3 over [0,2] using 4 subdivisions.

Here, for each interval we will approximate f by the value at the right endpoint. This gives multiple rectangles with base Q and height f(a + iQ). Doing this for i = 1, 2, ..., n−1, n, and adding up the resulting areas gives us

Q\left[f(a + Q) + f(a + 2Q)+\cdots+f(b)\right].\,

The right-hand Riemann sum will be an overestimation if the function f is monotonically increasing, and an underestimation if it is monotonically decreasing.

[edit] Middle sum

A middle Riemann sum of x3 over [0,2] using 4 subdivisions.

In this case we will take as approximation for f in each interval its value at the midpoint. For the first interval we will thus have f(a + Q/2), for the next one f(a + 3Q/2), and so on until f(b-Q/2) is reached. Summing up the areas, we find

Q\left[f(a + Q/2) + f(a + 3Q/2)+\cdots+f(b-Q/2)\right].

The error of this formula will be

\left \vert \int_{a}^{b} f(x) - A_\mathrm{mid} \right \vert \le \frac{M_2(b-a)^3}{(24n^2)},

where M2 is the maximum value of the absolute value of f^{\prime\prime}(x) on the interval.

[edit] Trapezoidal rule

A trapezoidal Riemann sum of x3 over [0,2] using 4 subdivisions.

In this case, the values of the function f on an interval will be approximated by the average of the values at the left and right endpoints. In the same manner as above, a simple calculation using the area formula A = h(b1 + b2) / 2 for a trapezium with parallel sides b1, b2 and height h one calculates the Riemann sum to be

\frac{1}{2}Q\left[f(a) + 2f(a+Q) + 2f(a+2Q) + 2f(a+3Q)+\cdots+f(b)\right].

The error of this approximation for the integral is

\left \vert \int_{a}^{b} f(x) - A_\mathrm{trap} \right \vert \le \frac{M_2(b-a)^3}{(12n^2)},

where M2 is the maximum value of the absolute value of f^{\prime\prime}(x).

[edit] Examples

[edit] Finding the area under the curve of y = x2

A visual representation of the area under the curve y = x2 for the interval from 0 to 2.

Suppose we have the function y = x2 and we want to find the area under the curve from 0 to 2. This looks like the graph to the right and we can use the right Riemann sum to find this area.

First we divide the interval from 0 to 2 into n subintervals, each of which will have a width of \frac{2}{n}. We can consider these to be the width of our rectangles. Since we are using the right Riemann sum, our sequence of x coordinates for our boxes will be x_1, x_2, \cdots, x_{n}. Therefore the sequence of the heights of the boxes will be (x_1)^2, (x_2)^2, (x_3)^2, \cdots, (x_{n})^2. Note that x_i = \frac{2i}{n}, and xn = 2.

The area of each box will be \frac{2}{n} \times (x_i)^2 and therefore the nth left Riemann sum will be :S = \frac{2}{n} \times \left(\frac{2}{n}\right)^2 + ... + \frac{2}{n} \times \left(\frac{2i}{n}\right)^2 + ... + \frac{2}{n} \times \left(2\right)^2. Hence:

S = \frac{8}{n^3} \left(1 + \cdots + i^2 + \cdots + n^2\right)
S = \frac{8}{n^3} \left(\frac{n(n+1)(2n+1)}{6}\right)
S = \frac{8}{n^3} \left(\frac{2n^3+3n^2+n}{6}\right)
S = \frac{8}{3} + \frac{4}{n} + \frac{4}{3n^2}

If we look at the limit as n \rightarrow \infty, we should expect that as we increase the number of boxes, our approximation approaches the actual value of the area under the curve. Hence:

S = \lim_{n \rightarrow \infty}\left(\frac{8}{3} + \frac{4}{n} + \frac{4}{3n^2}\right)
S = \frac{8}{3}

Which correspond to the definite integral:

\int_{0}^{2} \left(x^{2}\right) dx = \frac{8}{3}

[edit] See also

[edit] References

  • Thomas, George B. Jr.; Finney, Ross L. (1996), Calculus and Analytic Geometry (9th ed.), Addison Wesley, ISBN 0-201-53174-7 

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Riemann_sum"
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