Definite integral

Executive Summary:

This document reviews the "Definite Integral JavaScript Simulation Applet HTML5" provided by Open Educational Resources / Open Source Physics @ Singapore. The applet is designed as an interactive tool to demonstrate the concept of definite integration of the sine function using the method of summing approximative rectangles. It allows users to manipulate parameters such as the integration limits and the number of subintervals to visualize how the Riemann sum approaches the definite integral. This resource is intended for learning and teaching calculus, particularly the fundamental concept of definite integration.

Main Themes and Important Ideas/Facts:

1. Demonstration of Definite Integration: The core purpose of the applet is to visually illustrate the definition of a definite integral as the limit of a Riemann sum. It specifically focuses on the sine function. The description states: "This simulation demonstrates definite integration of the sine function by the simple algorithm of summing approximative rectangles."
2. Algorithm of Summing Approximative Rectangles: The simulation employs the fundamental method of approximating the area under a curve by dividing it into a series of rectangles. The height of each rectangle is determined by the function's value at the beginning of each subinterval. The text notes: "In each subinterval the approximative amplitude is assumed to be equal to its initial value. The sum of the area of all rectangles is shown by a green point at x2."
3. Interactive Parameters: The applet provides users with interactive controls to explore the concept:

• Initial Point (x1): Controlled by a blue draggable box and a "first slider." The description indicates: "A first slider defines x1."
• End Point (x2): Controlled by a magenta draggable box that can be drawn with the mouse. The text states: "x2 can be drawn with the mouse."
• Number of Subintervals (n): Controlled by a "second slider." The applet divides the interval [x1, x2] into n-1 subintervals for approximation. The description mentions: "A second slider n defines the number n-1 of subinterval into which x2 - x1 is divided for the approximation."

1. Visual Representation: The simulation provides multiple visual cues to aid understanding:

• Red Curve: Represents the sine function itself.
• Blue and Magenta Markers: Indicate the initial (x1) and end (x2) points of integration.
• Rectangles: Illustrate the approximative areas used in the Riemann sum.
• Green Point at x2: Represents the sum of the areas of all the rectangles.
• Analytic Antiderivative (Red Curve): Shows the exact value of the definite integral, calculated using the antiderivative of the sine function: y = cos(x) - cos (x1). The text explains: "The red curve is the analytic antiderivative for an initial value x1 : y = cos(x) - cos (x1) ."

1. Experiments for Guided Learning: The "Experiments" section suggests structured ways for users to interact with the simulation and develop their understanding:

• E1: Observing the deviation between the rectangle sum and the analytic solution with default settings.
• E2: Understanding why the deviation can be negative or zero, and how errors in approximation can compensate for periodic functions.
• E3: Observing how increasing the number of intervals leads to the deviation disappearing in the limit, reinforcing the definition of the definite integral.
• E4: Investigating how changing the initial point affects the analytic solution.
• E5: Considering why the rectangles might lie above or below the function in different parts of a large integration interval, relating to the increasing/decreasing nature of the function.

1. Instructions for Use: Clear instructions are provided for interacting with the applet:

• N Slider: Controls the number of approximation boxes.
• Draggable Boxes: Allow setting the integration limits (x1 and x2).
• Toggling Full Screen: Enabled by double-clicking.
• Reset Button: Returns the simulation to its default state (1 < x < 4 and n-1 = 9).

1. Context within a Larger Resource: The applet is part of a broader collection of "Open Educational Resources / Open Source Physics @ Singapore," which includes numerous other JavaScript simulations for learning mathematics and physics. The categorization under "Calculus" and "Learning and Teaching Mathematics using Simulations" highlights its intended educational use. The mention of "Plus 2000 Examples from Physics" suggests a rich set of related resources.
2. Technical Details and Licensing: The applet is implemented in HTML5 and utilizes JavaScript (likely the EasyJavaScriptSimulations Library, as indicated in the footer). The content is licensed under the Creative Commons Attribution-Share Alike 4.0 Singapore License for non-commercial use, with specific licensing terms for commercial use of the EJS library requiring direct contact.

Quotes from Original Source:

• "This simulation demonstrates definite integration of the sine function by the simple algorithm of summing approximative rectangles."
• "The definite integral is to be calculated between initial point x1 (blue) an end point x2(magenta)."
• "A first slider defines x1 ; x2 can be drawn with the mouse."
• "The red curve is the analytic antiderivative for an initial value x1 : y = cos(x) - cos (x1) ."
• "A second slider n defines the number n-1 of subinterval into which x2 - x1 is divided for the approximation."
• "In each subinterval the approximative amplitude is assumed to be equal to its initial value. The sum of the area of all rectangles is shown by a green point at x2."
• "E3: Increase the number of intervals with the slider and observe how the deviation changes and disappears in the limit."
• "Toggling the n slider will toggle the number of boxes used for approximation. (n=49)"
• "Drag the orange/magenta boxes to set the points/coordinates on the curve."
• "Reset defines 1 < x < 4 and n-1 = 9."

Conclusion:

The "Definite Integral JavaScript Simulation Applet HTML5" is a valuable open educational resource for teaching and learning the fundamental concept of definite integration. Its interactive nature, clear visual representations, and guided experiments allow users to actively explore how Riemann sums approximate the area under the sine curve and converge to the definite integral as the number of subintervals increases. The applet's inclusion within a larger collection of educational simulations further enhances its potential for integrated learning experiences in mathematics and physics.

Study Guide: Definite Integral Simulation Applet

Core Concepts

• Definite Integral: The definite integral represents the signed area between the graph of a function and the x-axis over a specified interval [x1, x2].
• Riemann Sum: The definite integral can be approximated by dividing the interval [x1, x2] into n subintervals and summing the areas of rectangles whose heights are determined by the function's value at some point within each subinterval. In this simulation, the height is taken at the initial value of each subinterval.
• Approximation by Rectangles: This simulation uses the left Riemann sum, where the height of each rectangle is the function value at the left endpoint of the subinterval.
• Number of Subintervals (n): Increasing the number of subintervals generally improves the accuracy of the Riemann sum approximation of the definite integral. As n approaches infinity, the Riemann sum approaches the exact value of the definite integral.
• Analytic Antiderivative: The Fundamental Theorem of Calculus provides a method to calculate the exact definite integral of a function if an antiderivative of that function is known. For the sine function, an antiderivative is the negative cosine function (or related forms, as shown in the simulation).
• Deviation: The difference between the approximate value of the definite integral (obtained by the Riemann sum) and the exact value (obtained using the analytic antiderivative).
• Periodic Functions: Functions that repeat their values at regular intervals. The sine function is a periodic function.

Using the Simulation

• x1 Slider (Blue): This slider allows you to set the initial point of the integration interval.
• x2 (Magenta): The end point of the integration interval, which can be dragged with the mouse.
• n Slider: This slider controls the number of subintervals (n-1) used for the rectangular approximation. Increasing n increases the number of rectangles.
• Red Curve: Represents the sine function, f(x) = sin(x).
• Green Point: Represents the sum of the areas of the approximating rectangles, plotted at the x2 value.
• Red Curve (Analytic Antiderivative): Shows the graph of y = cos(x) - cos(x1), which represents the exact definite integral of sin(t) from x1 to x.
• Reset Button: Returns the simulation to its default settings (1 < x < 4 and n-1 = 9).

Experiments to Understand

• Experiment E1: Observe how the green point (approximation) deviates from the red curve (exact integral) for the default settings.
• Experiment E2: Analyze why the approximation is less than the exact value when x < π/2 and becomes equal at x = π. Consider how errors in the rectangular approximation can sometimes cancel out over a periodic function.
• Experiment E3: Understand the relationship between the number of subintervals (n) and the accuracy of the approximation. Observe that as n increases, the deviation decreases.
• Experiment E4: Investigate how changing the initial point (x1) affects the analytic solution and why this shift occurs based on the properties of definite integrals.
• Experiment E5: Consider why, over large intervals, the rectangles might sometimes be below and sometimes above the sine function, relating this to the function's increasing and decreasing behavior.

Quiz

1. What mathematical concept does this simulation primarily demonstrate? Explain the basic idea behind this concept in 2-3 sentences.
2. Describe how the definite integral is approximated in this simulation. What shape is used for the approximation, and how is its size determined?
3. What does the 'n' slider control in the simulation, and how does changing its value typically affect the accuracy of the approximation? Explain your reasoning.
4. What is the analytic antiderivative shown in the simulation for the sine function, and how does it relate to the definite integral according to the Fundamental Theorem of Calculus?
5. Explain what the green point represents in the simulation and how its position is determined.
6. In Experiment E2, why is the deviation negative for x < π/2 when using the default approximation method?
7. According to Experiment E2, how can "summing mistakes" compensate for periodic functions when approximating a definite integral? Provide a brief explanation.
8. What happens to the deviation between the approximation and the analytic solution as the number of subintervals (n) is significantly increased (as suggested in Experiment E3)? Why does this occur?
9. How does changing the initial point of integration (x1) shift the analytic solution, as explored in Experiment E4? Relate this to a property of definite integrals.
10. In Experiment E5, why might the approximating rectangles lie below certain parts and above other parts of the sine function over a large integration interval?

Quiz Answer Key

1. This simulation demonstrates the concept of the definite integral. The definite integral represents the net signed area between a function's graph and the x-axis over a specific interval. It can be understood as the limit of a sum of areas of approximating rectangles.
2. The definite integral is approximated by summing the areas of rectangles. For each subinterval, a rectangle is constructed with a width equal to the subinterval's length and a height equal to the function's value at the left endpoint of that subinterval.
3. The 'n' slider controls the number of subintervals used to divide the integration interval. Increasing 'n' generally improves the accuracy of the approximation because the width of each rectangle becomes smaller, allowing the sum of the rectangular areas to more closely fit the area under the curve.
4. The analytic antiderivative shown is y = cos(x) - cos(x1). According to the Fundamental Theorem of Calculus, the definite integral of a function f(x) from a to b is equal to F(b) - F(a), where F(x) is an antiderivative of f(x). In this case, the antiderivative of sin(x) is -cos(x), leading to the form shown with the initial value x1.
5. The green point represents the cumulative sum of the areas of all the approximating rectangles from x1 up to the current value of x2. Its vertical position corresponds to this total area, plotted at the x-coordinate x2.
6. For x < π/2, the sine function is increasing. When using the left Riemann sum (where the height of the rectangle is the function value at the left endpoint), the rectangles' heights are slightly less than the function's value over most of the subinterval, leading to an underestimation of the area and thus a negative deviation.
7. For periodic functions like sine, over a complete cycle or parts thereof, an underestimation of the area in one region might be compensated by an overestimation in another region when using a rectangular approximation. This can lead to a smaller overall deviation at specific points, as observed at x = π.
8. As the number of subintervals (n) increases, the deviation between the approximation and the analytic solution tends to disappear. This happens because the width of each approximating rectangle becomes infinitesimally small, and the sum of their areas converges to the exact area under the curve, which is given by the definite integral.
9. Changing the initial point of integration (x1) shifts the constant term in the analytic solution. This is because the definite integral calculates the area relative to the starting point. A different starting point implies a different accumulation of area.
10. Over a large interval, the sine function increases and decreases. In regions where the function is increasing, the left Riemann sum will generally underestimate the area. Conversely, in regions where the function is decreasing, the left Riemann sum will generally overestimate the area. This leads to the rectangles lying below or above different parts of the curve.

Essay Format Questions

1. Discuss the relationship between the definite integral and the approximation by Riemann sums, as illustrated by the JavaScript simulation. Explain how the accuracy of the approximation changes with the number of subintervals and connect this to the formal definition of the definite integral.
2. Analyze the behavior of the deviation between the rectangular approximation and the analytic solution for the sine function, considering its periodic nature. Discuss why the deviation might be positive, negative, or zero at different endpoints of the integration interval.
3. Explain the significance of the analytic antiderivative in the context of definite integrals, referencing the Fundamental Theorem of Calculus. Describe how the simulation visually represents this relationship and how changing the initial point of integration affects the analytic solution.
4. Critically evaluate the use of this JavaScript simulation as a tool for learning about definite integrals. Discuss its strengths and limitations in helping students understand the underlying concepts and the process of approximation.
5. Based on your observations from the experiments suggested with the simulation, discuss the factors that influence the accuracy of approximating a definite integral using Riemann sums. Consider the properties of the function being integrated and the parameters of the approximation method.

Glossary of Key Terms

• Definite Integral: A numerical value representing the signed area between the graph of a function and the x-axis over a specified interval.
• Riemann Sum: An approximation of the definite integral calculated by summing the areas of rectangles that approximate the region under the curve.
• Subinterval: One of the smaller intervals into which the integration interval is divided when calculating a Riemann sum.
• Left Riemann Sum: A type of Riemann sum where the height of each approximating rectangle is the function value at the left endpoint of its subinterval.
• Analytic Antiderivative: A function whose derivative is the original function. It is used to calculate the exact value of a definite integral.
• Fundamental Theorem of Calculus: A theorem that links the concept of the derivative of a function with the concept of the integral. It states that the definite integral of a function can be calculated using its antiderivative.
• Deviation (in this context): The difference between the approximate value of the definite integral (from the Riemann sum) and the exact value (from the analytic antiderivative).
• Periodic Function: A function that repeats its values at regular intervals; for example, the sine function.
• Integration Interval: The interval [x1, x2] over which the definite integral is being calculated.
• Approximation: An estimate or inexact calculation of a value. In this case, the Riemann sum is an approximation of the definite integral.