Worked example: Rewriting definite integral as limit of Riemann sum | AP Calculus AB | Khan Academy

Khan Academy
1 Aug 201705:34
EducationalLearning
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TLDRThe video script presents a detailed walkthrough on how to rewrite a definite integral as the limit of a Riemann sum. The example used is the integral of the cosine function from pi to two pi. The explanation includes dividing the interval into n equal parts, calculating the width of each rectangle, and determining the height based on the value of the function at the right endpoint. The process culminates in expressing the integral as the limit of the sum of the areas of these rectangles as n approaches infinity, providing a clear understanding of the Riemann sum concept.

Takeaways
  • 📚 The concept being discussed is the practice of rewriting definite integrals as the limit of a Riemann sum.
  • 🌟 The specific example given is the definite integral from pi to two pi of the cosine of x dx.
  • 🎯 The goal is to express this integral in the form of a limit as n approaches infinity using Riemann sums.
  • 📊 Riemann sums are visualized by breaking up the integral into rectangles, with each rectangle representing a small part of the whole area under the curve.
  • 📈 The width of each rectangle is determined by dividing the interval from pi to two pi by the number of rectangles, n, resulting in a width of pi over n.
  • 📉 For a right Riemann sum, the height of each rectangle is defined by the value of the function at the right endpoint of the rectangle.
  • 🌠 The general form of the height for the i-th rectangle is the cosine of (pi plus pi over n times i).
  • 🔢 The final expression for the Riemann sum involves the sum of the areas of the rectangles, which is the product of the height and the width for each rectangle.
  • 🚀 As n approaches infinity, the Riemann sum more accurately approximates the area under the curve, eventually equaling the value of the definite integral.
  • 🧩 The process illustrates how definite integrals can be conceptualized and calculated through the limit of Riemann sums, providing a geometric interpretation of integration.
  • ✨ The exercise reinforces the understanding of the fundamental theorem of calculus and the relationship between integrals and their Riemann sum representations.
Q & A

  • What is the definite integral being discussed in the script?

    -The definite integral discussed is from pi to two pi of the cosine of x with respect to dx.

  • How is the definite integral rewritten in the context of the script?

    -The definite integral is rewritten as the limit as n approaches infinity of a Riemann sum.

  • What is the Riemann sum representing in this context?

    -The Riemann sum represents the sum of the areas of n rectangles, which approximate the area under the curve of the cosine function from pi to two pi.

  • What is the value of the definite integral in this case?

    -The value of the definite integral in this case is zero, as the positive and negative areas under the curve cancel each other out.

  • How is the interval from pi to two pi divided for the Riemann sum?

    -The interval is divided into n equal intervals, with each interval having a width of pi over n.

  • What determines the height of each rectangle in the Riemann sum?

    -The height of each rectangle is determined by the value of the cosine function at the right endpoint of the interval.

  • What is the general form of the height for the i-th rectangle?

    -The general form of the height for the i-th rectangle is cosine of (pi plus pi over n times i).

  • What is the purpose of rewriting a definite integral as a Riemann sum?

    -Rewriting a definite integral as a Riemann sum helps in approximating the integral using discrete sums, which can be useful for computation and understanding the accumulation of areas under a curve.

  • What is the significance of the limit as n approaches infinity in this context?

    -The limit as n approaches infinity signifies that the approximation of the Riemann sum becomes more accurate, as the rectangles more closely approximate the actual area under the curve.

  • How does the graph of the cosine function contribute to the understanding of this problem?

    -The graph of the cosine function helps visualize the areas under consideration and how they combine to form the definite integral, with the negative and positive areas canceling each other out.

  • What is the role of the Sigma notation in expressing the Riemann sum?

    -The Sigma notation is used to express the Riemann sum as a mathematical series, summing up the areas of the individual rectangles to approximate the definite integral.

Outlines
00:00
📚 Introduction to Riemann Sums

The paragraph begins with an instructor introducing the concept of Riemann sums as a method to rewrite definite integrals. The example given is the definite integral from pi to two pi of the cosine of x dx. The goal is to express this integral as the limit as n approaches infinity of a Riemann sum. The instructor explains the process of breaking the integral into rectangles, where the height of each rectangle is determined by the value of the function at the right boundary of the rectangle. The width of each rectangle is calculated by dividing the interval from pi to two pi by n, resulting in pi over n. The paragraph ends with the instructor drawing a visual representation to help understand the concept better.

05:05
📈 Calculation of Riemann Sums

In this paragraph, the instructor continues the explanation of Riemann sums by detailing the calculation process. The focus is on the height of each rectangle, which is determined by the function value at the right boundary. The general form of the right boundary for the i-th rectangle is explained as pi plus pi over n times i. The width of the rectangles, already calculated in the previous paragraph, is reiterated as pi over n. The paragraph concludes with the instructor expressing that the definite integral has been successfully re-expressed as the limit of a right Riemann sum, providing a clear understanding of the relationship between Riemann sums and definite integrals.

Mindmap
Keywords
💡Definite Integral
A definite integral represents the signed area under a curve between two points on the x-axis. In the video, the definite integral from pi to two pi of the cosine of x dx is discussed, which calculates the area between the curve of the cosine function and the x-axis over the specified interval.
💡Limit
In mathematics, a limit is the value that a function or sequence approaches as the input (or index) approaches some value. The video discusses the concept of rewriting a definite integral as the limit of a Riemann sum as n approaches infinity, which means as the number of rectangles used to approximate the integral increases without bound.
💡Riemann Sum
A Riemann sum is a method used to approximate the definite integral of a function by dividing the area under the curve into small rectangles and summing their areas. The video focuses on the right Riemann sum, where the height of each rectangle is determined by the function's value at the right endpoint of each interval.
💡Sigma Notation
Sigma notation, also known as summation notation, is a way of expressing the sum of a sequence or series mathematically. In the context of the video, it is used to represent the sum of the areas of the rectangles that approximate the Riemann sum.
💡Function
A function is a mathematical relation that associates each element from one set (called the domain) to exactly one element of another set (called the range). In the video, the function being integrated is the cosine function, and its values at different points are used to determine the height of the rectangles in the Riemann sum.
💡Interval
In mathematics, an interval is a set of real numbers lying between two numbers, including or excluding the numbers themselves. The video discusses dividing the interval from pi to two pi into n equal subintervals to approximate the definite integral using Riemann sums.
💡Width
In the context of the Riemann sum, the width refers to the length of the base of each rectangle used in the approximation. The width is calculated by dividing the length of the interval (two pi minus pi) by the number of rectangles (n).
💡Height
The height in the context of the Riemann sum is the value of the function at the right endpoint of each rectangle, which determines the height of the rectangle used in the approximation of the integral.
💡Cosine Function
The cosine function is a fundamental trigonometric function that describes the periodic behavior of waves. In the video, the cosine function is integrated over the interval from pi to two pi, and its graph is used to visualize the area under consideration.
💡Graph
A graph is a visual representation of the relationship between variables, often used to plot the values of a function. In the video, the graph of the cosine function is drawn to help understand the area represented by the definite integral and the Riemann sum approximation.
💡Approximation
Approximation in mathematics is the process of finding a value that is close to the actual value but easier to work with or understand. The video demonstrates the approximation of a definite integral using Riemann sums, which become more accurate as the number of rectangles (n) increases.
Highlights

Rewriting definite integrals as the limit of a Riemann sum is a fundamental concept in calculus.

The definite integral from pi to two pi of cosine of x dx is used as an example to illustrate the process.

The limit as n approaches infinity is a key aspect of the Riemann sum definition.

The use of Sigma notation is essential in expressing the Riemann sum mathematically.

The graph of cosine of x is drawn to visually represent the area under consideration.

The area represented by the definite integral from pi to two pi is zero due to the cancellation of positive and negative parts.

Breaking the interval into n rectangles is a step in approximating the definite integral using Riemann sums.

The right boundary of each rectangle defines the height in a right Riemann sum.

The width of each rectangle is calculated by dividing the interval length by n.

The height of each rectangle is determined by the function value at the right boundary.

The general form for the height of the i-th rectangle is provided, which involves the function value at (pi + pi/n * i).

The process of rewriting the integral as a Riemann sum involves understanding the behavior of the function within the interval.

The limit of the sum of the areas of the rectangles as n approaches infinity is the value of the definite integral.

This method of rewriting integrals provides a deeper understanding of the Riemann sum and its approximation to the true integral.

The practical application of Riemann sums can be seen in numerical methods for solving definite integrals.

The transcript provides a clear and detailed explanation of the Riemann sum concept, suitable for educational purposes.

Transcripts

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