I - Riemann Sums and Numerical Integration Lesson

Riemann Sums and Numerical Integration

Riemann Sums

image of Reimann The rectangle approximation methods used to find areas involved summing areas of rectangles of equal width. These were special cases of a more general sum where it is not necessary to have subintervals of equal width.

image of right hand versus left hand graphs

Let f be defined on the closed interval [a, b], and let Δ be a partition of [a, b] given by a = x₀ < x₁ < x₂ < . . . < x_n-1 < x_n = b

where equation image indicator is the width of the ith subinterval. If c_i is any point in the ith subinterval, then the sum $LaTeX: \sum_{i=1}^{n}=f(c_{i}) \Lambda x_{i}$ $\sum_{i=1}^{n}=f(c_{i}) \Lambda x_{i}$ , where x_i-1 < c_i < x_i is a Riemann sum of f for the partition $LaTeX: \Delta$ $\Delta$ If f(c_i) is positive, the number equation image indicator $LaTeX: f(c_{i}) \Lambda x_{i}$ $f(c_{i}) \Lambda x_{i}$ = height * base is the area of the rectangle. If f(c_i) is negative, then $LaTeX: f(c_{i}) \Lambda x_{i}$ $f(c_{i}) \Lambda x_{i}$ is the negative of the area. In general, if the function f takes on both positive and negative values, then the Riemann sum is the sum of the areas of the rectangles lying above the x-axis and the negative of the areas of the rectangles lying below the x-axis.

The width of the largest subinterval of a partition Δ is the norm of the partition and is denoted by equation image indicator $LaTeX: \|\Delta \|$ $\|\Delta \|$ . If every subinterval is of equal width, the partition is denoted by $LaTeX: \|\Delta\| =\Delta x=\frac{b-a}{n}$ $\|\Delta\| =\Delta x=\frac{b-a}{n}$ . For a general partition, the norm is related to the number of subintervals of [a, b] by equation image indicator $LaTeX: \frac{b-a}{\|\Delta\|}\le n$ $\frac{b-a}{\|\Delta\|}\le n$ . The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0, i.e., $LaTeX: \|\Delta \|\to 0$ $\|\Delta \|\to 0$ implies that $LaTeX: n \to \infty$ $n \to \infty$ .

Links to an external site.Explore the midpoint and trapezoidal Reimann sum by CLICKING HERE. Links to an external site.

Reimann image caption: The German mathematician Georg Friedrich Bernhard Riemann (1826-1866) made lasting contributions to analysis and differential geometry, some of them enabling the later development of Einstein\'s theory of general relativity.

Numerical Integration

There are times when it is extremely difficult or impossible to find the exact value of a definite integral. This may occur when a definite integral involving an elementary function does not have an elementary function for its antiderivative or when the function to be integrated is determined from a scientific experiment's instrument readings or collected data and no formula for the function is apparent. In situations such as these, numerical approximation techniques offer a viable alternative. Numerical integration is the process of approximating the value of an integral (area under a curve) using numerical techniques when it is not possible to find a formula that can be evaluated to give the value of a definite integral.

Trapezoidal Rule

The Trapezoidal Rule is a relatively simple numerical technique for estimating an integral's value. It approximates short sections of a curve with line segments and sums the areas of the trapezoids formed by joining the ends of these segments to the x- axis. Click HERE to view the presentation on how such trapezoidal sums form the basis for the Trapezoidal Rule. Links to an external site.

View the video below on the Trapezoidal Rule.

Riemann Sums Practice

1. Find the norm of the partition equation image indicator $LaTeX: \Delta=\left\{-2,\:-1.7,\:-0.5,\:0,\:0.9,\:1\right\}$ $\Delta=\left\{-2,\:-1.7,\:-0.5,\:0,\:0.9,\:1\right\}$ .

Solution: 1.2 text annotation indicator

2. Use RRAM6 to approximate the area under f(x) = 1/x on the interval [2, 4] to the nearest hundredth.

Solution: .653 text annotation indicator

3. Use MRAM6 to approximate the area under f(x) = 4x + 3 on the interval [5, 8] to the nearest hundredth.

Solution: 87 text annotation indicator

4. Use the table and LRAM5 as divided by the table to estimate the area under the curve.

Solution: 605

x	0	5	10	20	30	35
f(x)	0	5	20	30	16	0

Numerical Integration Practice

1. Use the Trapezoidal Rule with n = 5 to approximate the integral equation image indicator $LaTeX: \int_1^{2}\frac{1}{x}dx$ $\int_1^{2}\frac{1}{x}dx$ to 6 decimal places.

Solution: text annotation indicator 0.695635

2. Use the Trapezoidal Rule with n = 8 to approximate the integral equation image indicator $LaTeX: \int_1^{5}\frac{\cos x}{x}dx$ $\int_1^{5}\frac{\cos x}{x}dx$ to 6 decimal places.

Solution: -0.495333 text annotation indicator

Riemann Sums: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of Riemann Sums and Numerical Integration.

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