Riemann sum | MIT 18.01SC Single Variable Calculus, Fall 2010

MIT OpenCourseWare
7 Jan 201107:25
EducationalLearning
32 Likes 10 Comments

TLDRIn this recitation, the professor introduces the concept of using Riemann sums to approximate definite integrals, specifically the integral of x cubed from -1 to 3. The lesson involves dividing the interval into four equal subintervals and using left endpoints to estimate the integral's value. The professor visually demonstrates the process by drawing rectangles on a graph, calculating the area of each, and summing them up to find an approximation. The final step includes a formal representation of the Riemann sum and an explanation of the significance of each component in the calculation.

Takeaways
  • πŸ‘‹ The professor introduces the session focused on using Riemann sums to approximate definite integrals.
  • πŸ“ The specific problem given is to use four subintervals and left endpoints to approximate the integral of x cubed from -1 to 3.
  • πŸ“Š A rough sketch of y = x cubed is provided to aid in understanding.
  • πŸ•’ The professor gives students time to work on the problem before demonstrating the solution.
  • πŸ“ Four subintervals with left endpoints are used to approximate the integral.
  • πŸ“‰ The heights of the rectangles are determined by the output at the left endpoint of each subinterval.
  • πŸ”’ The values of y_0, y_1, y_2, and y_3 are calculated based on the function x cubed.
  • βž— The left endpoints are at x = -1, 0, 1, and 2, resulting in y values of -1, 0, 1, and 8 respectively.
  • 🟦 The areas of the rectangles are summed to approximate the definite integral, resulting in a total area of 8.
  • πŸ”’ The Riemann sum is written formally as the sum of f(x_i) * delta x from i = 0 to 3.
Q & A

  • What is the main topic of the recitation session described in the transcript?

    -The main topic is using Riemann sums to approximate definite integrals.

  • What is the function given in the problem to be integrated?

    -The function given is f(x) = x^3, or y = x cubed.

  • Over what interval is the definite integral to be approximated?

    -The definite integral is to be approximated over the interval from -1 to 3.

  • How many subintervals are used to approximate the definite integral in the problem?

    -Four subintervals are used for the approximation.

  • What method of Riemann sums is being used in the approximation?

    -The left endpoint method of Riemann sums is being used.

  • What is the value of delta x in this problem?

    -In this problem, delta x is equal to 1, as each subinterval has a unit length.

  • What are the left endpoints of the subintervals used in the approximation?

    -The left endpoints are -1, 0, 1, and 2.

  • How are the heights of the rectangles determined in the left endpoint method?

    -The heights of the rectangles are determined by the function values at the left endpoints of the subintervals.

  • What is the value of y_0 in the approximation?

    -The value of y_0 is the function value at x = -1, which is (-1)^3 or -1.

  • What is the sum of the areas of the rectangles used to approximate the definite integral?

    -The sum of the areas is -1 + 0 + 1 + 8, which equals 8.

  • How is the Riemann sum represented in the transcript?

    -The Riemann sum is represented as the sum from i = 0 to 3 of f(x_i) * delta x.

  • What is the significance of the negative sign in the area calculation?

    -The negative sign indicates that the area below the x-axis is considered as negative when calculating the signed area under the curve.

  • How can the Riemann sum be written more formally with the x_i values?

    -The Riemann sum can be written formally as the sum from i = 0 to 3 of f(x_i), where x_0 = -1, x_1 = 0, x_2 = 1, and x_3 = 2, each multiplied by delta x.

Outlines
00:00
πŸ“š Introduction to Riemann Sums for Definite Integrals

The professor begins the recitation by introducing the concept of using Riemann sums to approximate definite integrals. The task is to approximate the integral of the function x^3 from -1 to 3 using four subintervals with left endpoints. The students are given time to work on the problem, which involves sketching the function and understanding the concept of left endpoint rectangles for approximation. The professor also explains the importance of considering the signed area under the curve and how to calculate the area of the rectangles formed by the left endpoints.

05:04
πŸ“ Calculating Riemann Sums with Left Endpoints

In this paragraph, the professor demonstrates the calculation of the Riemann sum using the left endpoints of the subintervals. The function f(x) = x^3 is evaluated at the left endpoints x = -1, 0, 1, and 2, yielding values y_0 = -1, y_1 = 0, y_2 = 1, and y_3 = 8. The professor emphasizes the importance of the subscripts representing the interval rather than the x-value. The rectangles' areas are calculated by multiplying the base (delta x, which is 1) by the respective heights (y_values). The sum of these areas, considering the negative area of the first rectangle, results in an approximation of the definite integral. The professor also introduces the formal notation for Riemann sums and how to represent the x-values for each subinterval.

Mindmap
Keywords
πŸ’‘Riemann sums
Riemann sums are a method for approximating the total area under a curve, often used to estimate definite integrals. In the video, the professor explains how to use Riemann sums with four subintervals and left endpoints to approximate the integral of the function x^3 over a given interval.
πŸ’‘Definite integral
A definite integral represents the signed area under a curve within a specified interval. It provides the net accumulation of a quantity. The video focuses on approximating the definite integral of the function x^3 from -1 to 3 using Riemann sums.
πŸ’‘Left endpoints
Left endpoints refer to the points at the left side of each subinterval used in calculating Riemann sums. The height of each rectangle in the Riemann sum approximation is determined by the function value at these left endpoints. In the video, the left endpoints are used to approximate the integral of x^3.
πŸ’‘Subintervals
Subintervals are divisions of the main interval over which the Riemann sum is calculated. In the video, the interval from -1 to 3 is divided into four subintervals, each of unit length, to approximate the integral using left endpoints.
πŸ’‘Function x^3
The function x^3 represents a cubic function used as an example in the video to demonstrate Riemann sums. The integral of this function over a specified interval is approximated by calculating the areas of rectangles under the curve using left endpoints.
πŸ’‘Signed area
Signed area is the concept of assigning positive or negative values to areas under a curve, depending on whether they are above or below the x-axis. In the video, this concept is important as it affects the calculation of the integral for the function x^3, especially where the curve dips below the x-axis.
πŸ’‘Delta x
Delta x (Ξ”x) represents the width of each subinterval in the Riemann sum approximation. In the video, delta x is consistently 1, as the interval from -1 to 3 is divided into four equal subintervals, each of length 1.
πŸ’‘Rectangle heights
Rectangle heights in the context of Riemann sums are determined by the function values at specific points (such as left endpoints) in each subinterval. These heights, multiplied by the base (delta x), give the area of each rectangle. In the video, the heights are calculated for the function x^3 at each left endpoint.
πŸ’‘Output values
Output values refer to the function values at specific input points, such as the left endpoints of subintervals. These values determine the height of rectangles in the Riemann sum approximation. The video illustrates how to find these values for the function x^3 to estimate the integral.
πŸ’‘Sum notation
Sum notation, often represented by the sigma symbol (Ξ£), is used to succinctly express the sum of multiple terms. In the video, sum notation is used to represent the Riemann sum formula, which adds up the areas of the rectangles formed under the curve using left endpoints and delta x.
Highlights

Introduction to using Riemann sums to approximate definite integrals.

Problem statement: Approximate the integral of x^3 from -1 to 3 using 4 subintervals and left endpoints.

Sketch of y = x^3 provided as a starting point.

Explanation of the process to approximate the definite integral using Riemann sums with left endpoints.

Drawing of the 4 subintervals and left endpoints on the graph.

Calculation of the actual value of the estimate using the heights of the rectangles at the left endpoints.

Introduction to different notations for writing Riemann sums.

Demonstration of how to calculate the height of each rectangle using the function's output at the left endpoints.

Evaluation of y_0 to y_3 for the heights of the rectangles.

Explanation of the signed area concept and its application in the approximation.

Calculation of the area of each rectangle and the overall approximation.

Formal representation of the Riemann sum using a summation notation.

Alternative way to write the Riemann sum with designated x values for each subinterval.

Clarification of the subscripts representing intervals rather than x-values.

Final calculation of the Riemann sum approximation: -1 + 1 + 8.

Visual representation of the rectangles and their areas on the graph.

Summary of the process and the result of using Riemann sums to approximate the definite integral of x^3.

Transcripts

Browse More Related Video

AP CALCULUS AB 2022 Exam Full Solution FRQ#4c

Business Calculus - Math 1329 - Section 5.3 - The Definite Integral and Fundamental Thm of Calculus

Riemann and Trapezoidal Sums from Tables of Values

AP Calculus AB: Lesson 6.2 Part 2 (Limit Definition of Definite Integral)

Left, Right, & Midpoint Riemann Sum Formulas

BusCalc 13.4 Riemann Sums

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
Riemann SumsIntegral ApproximationCalculus TutorialMath EducationDefinite IntegralsLeft EndpointsArea Under CurveCubed FunctionEducational ContentMath Techniques