Worked example: finding a Riemann sum using a table | AP Calculus AB | Khan Academy

Khan Academy
8 Sept 201706:39
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TLDRThe video script discusses the concept of approximating the area under a curve using a right Riemann sum with three equal subdivisions. It explains that even without the graph of the function, one can estimate the area by using the function's values at specific points. The process involves dividing the interval from x=1 to x=10 into three sections and using the function values at the right boundary of each section to determine the height of the rectangles that approximate the area. The script concludes by emphasizing that the accuracy of the approximation depends on the actual shape of the function, which is unknown.

Takeaways
  • 📊 The task involves approximating the area under a curve using a right Riemann sum with three equal subdivisions from x=1 to x=10.
  • 📈 Given a table of values for a function without its graph, the right Riemann sum can still be used to estimate the area.
  • 🔍 The approximation relies on the values of the function at specific points and does not require knowledge of the function's exact shape.
  • 🎯 The key points for approximation are the function values at x=1, x=2, x=3, x=4, ..., x=10.
  • 📐 The right Riemann sum divides the x-axis from 1 to 10 into three equal parts of width 3 units each.
  • 🏢 For each subdivision, the height of the rectangle is determined by the function value at the right boundary of that subdivision.
  • 📂 The first rectangle's height is given by f(4)=8, the second by f(7)=3, and the third by f(10)=5.
  • 🧮 The area of each rectangle is calculated as width times height: 3 units wide and the respective function values as heights.
  • 📈 The total approximated area is the sum of the areas of the three rectangles: 24 + 9 + 15, which equals 48 square units.
  • 🤔 The accuracy of the approximation depends on the true shape of the function, which could range from a good fit to a poor one.
  • 🔑 Despite the limitations, the right Riemann sum provides a practical method for estimating areas using discrete function values.
Q & A

  • What is the main topic of the video script?

    -The main topic of the video script is approximating the area under a curve using a right Riemann sum with three equal subdivisions.

  • What is the function f(x) in the context of the video?

    -The specific function f(x) is not given in the video script. The focus is on using the values of f(x) at certain points to approximate the area under the curve.

  • How many subdivisions are used in the right Riemann sum?

    -Three equal subdivisions are used in the right Riemann sum.

  • What are the x-values for each subdivision in the approximation?

    -The x-values for each subdivision are from x=1 to x=4, x=4 to x=7, and x=7 to x=10.

  • What is the value of f(x) at x=1, x=4, and x=7 according to the video?

    -The values are f(1) = 6, f(4) = 8, and f(7) = 3.

  • How is the height of the rectangles determined in a right Riemann sum?

    -The height of the rectangles in a right Riemann sum is determined by the value of the function at the right boundary of each subdivision.

  • What is the area of the first rectangle in the approximation?

    -The area of the first rectangle is 24 square units, calculated as 3 (width) times 8 (height).

  • What is the area of the second rectangle in the approximation?

    -The area of the second rectangle is 9 square units, calculated as 3 (width) times 3 (height).

  • What is the area of the third rectangle in the approximation?

    -The area of the third rectangle is 15 square units, calculated as 3 (width) times 5 (height).

  • What is the total approximated area under the curve using the right Riemann sum with three subdivisions?

    -The total approximated area is 48 square units, calculated as the sum of the areas of the three rectangles (24 + 9 + 15).

  • How does the accuracy of the approximation depend on the actual function f(x)?

    -The accuracy of the approximation depends on how closely the function f(x) behaves to a linear model within the subdivisions. If the function is very wiggly or complex, the Riemann sum may provide a less accurate approximation.

Outlines
00:00
📊 Understanding Right Riemann Sum for Area Approximation

This paragraph introduces the concept of using a right Riemann sum to approximate the area under a curve. The tutor explains that even without the entire graph of the function, one can approximate the area between the x-axis and the graph of a function, f, from x=1 to x=10 with three equal subdivisions. The process involves using a table of function values at certain points and visualizing the Riemann sums as rectangles, each representing a subdivision. The tutor emphasizes that the accuracy of the approximation depends on the actual shape of the function, which is unknown. The method is demonstrated by marking the points from the table on a graph and using the right boundary values of each subdivision to define the height of the rectangles. The approximation is then found by summing the areas of these rectangles.

05:01
📐 Calculating Area with Right Riemann Sum Rectangles

In this paragraph, the tutor continues the explanation of the right Riemann sum by detailing the calculation of the area approximation. The tutor describes how to calculate the area of each rectangle formed by the subdivisions, using the function values at the right boundary of each section. The first rectangle's area is calculated as 3 units wide by the height of f(4) which is 8, resulting in 24 square units. The second rectangle's area is 3 units wide by the height of f(7) which is 3, resulting in 9 square units. The third rectangle's area is 3 units wide by the height of f(10) which is 5, resulting in 15 square units. The total approximated area under the curve is then found by summing the areas of these three rectangles, which gives a total of 48 square units. The tutor reiterates that while this method provides an approximation, the actual accuracy is contingent on the true form of the function, which could vary significantly.

Mindmap
Keywords
💡Approximate the area
The process of estimating the size of a shape or area, in this case, the area under a curve, using mathematical methods. In the video, the tutor is guiding the viewer on how to approximate the area between the x-axis and the graph of a function from x=1 to x=10. The method used is the right Riemann sum, which involves dividing the area into smaller, manageable parts and calculating the area of these parts to estimate the total area.
💡Right Riemann sum
A method used in calculus for approximating the definite integral of a function. It involves dividing the interval under consideration into subintervals and using the function's value at the right endpoint of each subinterval to calculate the area of a rectangle that approximates the area under the curve. In the context of the video, the right Riemann sum is used to approximate the area under the graph of the function f(x) between x=1 and x=10 by calculating the area of rectangles formed using the function values at the right endpoints of the subdivisions.
💡Table of values
A collection of data points that represent the output of a function for a given set of input values. In the video, the table of values for the function f(x) provides the necessary information to perform the right Riemann sum. The values from the table are used to determine the height of the rectangles in the approximation process, with each value corresponding to the height of a rectangle for a specific subinterval.
💡Subdivisions
The act of dividing a larger interval or area into smaller parts. In the video, the interval from x=1 to x=10 is subdivided into three equal parts to facilitate the approximation of the area under the curve using the right Riemann sum method. Each subdivision is used to calculate the area of a rectangle, which contributes to the total area approximation.
💡Function values
The result of substituting a value into a function. In the context of the video, the function values are given for specific x-values and are used to approximate the area under the curve. These values are crucial for the right Riemann sum method as they determine the height of the rectangles that are used to estimate the area.
💡Rectangles
A geometric figure with four right angles. In the approximation of the area under a curve using the right Riemann sum, rectangles are used to estimate the area of the curve. The height of each rectangle corresponds to the function value at the right endpoint of a subdivision, and the width is the length of the subdivision. The area of each rectangle is then calculated and summed to approximate the total area under the curve.
💡Graph
A visual representation of the relationship between variables, often used in mathematics to depict functions. In the video, the graph of the function f(x) is not fully known, but the tutor uses the given function values to approximate its shape and the area under it. The graph is essential for visualizing the process of approximating the area, even though only certain points are known, and it helps to illustrate how the right Riemann sum method divides the area into rectangles.
💡Units
A standard of measurement used in various fields. In the context of the video, units refer to the standard of measurement for the area being approximated. The units are not specified but are implied when the tutor mentions 'square units,' indicating that the area is being measured in terms of the square of the unit of length used for the x-axis and y-axis.
💡Estimation
The process of making an educated guess or approximate calculation, often when exact values are unknown or difficult to determine. In the video, estimation is the main goal, as the tutor guides the viewer through the process of estimating the area under a curve using the right Riemann sum. Estimation is crucial in this context because the function's graph is not fully visible, and the tutor relies on the given function values to approximate the area.
💡Visualization
The act of creating a mental or graphical representation of an object, concept, or process. In the video, visualization is important for understanding the right Riemann sum method and how it approximates the area under a curve. The tutor emphasizes the importance of visualizing the Riemann sums by drawing axes and marking points, even though the exact shape of the function's graph is not known. This helps the viewer grasp the concept of dividing the area into rectangles and estimating the total area.
Highlights

The task involves approximating the area between the x-axis and a function's graph using a right Riemann sum with three equal subdivisions.

The approximation is done without the actual graph of the function, relying solely on the given values of the function at certain points.

The right Riemann sum is a method for approximating the area under a curve by dividing the region into rectangles.

The first step is to visualize the problem by drawing axes and marking the given points on the graph.

The given points are (1,6), (4,8), and (7,3) for the respective x-values.

The right Riemann sum uses the right boundary of each subdivision to define the height of the rectangles.

The first rectangle's height is determined by the function value at x=4, which is f(4)=8.

The second rectangle's height is determined by the function value at x=7, which is f(7)=3.

The third rectangle's height is determined by the function value at x=10, which is f(10)=5.

The area of each rectangle is calculated by multiplying the width (3 units) by the height (the function value at the right boundary).

The total approximation of the area is the sum of the areas of the three rectangles.

The calculated areas are 24 square units for the first rectangle, 9 square units for the second, and 15 square units for the third.

The final approximation of the area between x=1 and x=10 is 48 square units.

The accuracy of the approximation depends on the actual shape of the function between the given points.

The method demonstrates the practical application of right Riemann sums in approximating areas under curves without the need for the complete function graph.

Transcripts

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MathematicsCalculusRiemann SumsArea ApproximationFunction ValuesGraphical InterpretationSubdivisionRectanglesEducational ContentMath Tutorial