Defining Double Integration with Riemann Sums | Volume under a Surface

Dr. Trefor Bazett
1 Dec 201909:40
EducationalLearning
32 Likes 10 Comments

TLDRThis video explains the concept of finding the area under a surface in multivariable calculus, analogous to finding the area under a curve in single-variable calculus. It breaks down the process of subdividing a region into smaller rectangles (or boxes) to approximate the volume beneath a surface. The video demonstrates how to compute the area by taking the limit as the subdivisions become infinitely small, using an example function to illustrate the method. This approach, known as Riemann integration, allows for increasingly accurate approximations of the volume under a surface.

Takeaways
  • πŸ“š The concept of finding the area under a curve in multivariable calculus is analogous to single-variable calculus, involving breaking the region into small subdivisions to approximate the area.
  • πŸ“ In multivariable calculus, instead of rectangles, 'little boxes' are used to approximate the volume under a surface by summing up the volumes of these boxes.
  • πŸ” The approximation gets better with a larger number of subdivisions, making the boxes smaller and thus reducing the error in the approximation.
  • πŸ“ The height of each box is determined by plugging a specific point (chosen from within the box) into the function to get the function's value at that point.
  • πŸ”‘ The choice of the point within the box (XK, YK) for calculating the height does not matter in the limit as the number of subdivisions approaches infinity.
  • πŸ“ˆ The volume of each box is calculated as the product of the base area (Delta X * Delta Y) and the height (function value at XK, YK).
  • βˆ‘ The total volume under the surface is approximated by summing the volumes of all the small boxes within the partitioned region.
  • πŸš€ The Riemann integral defines the exact volume under the surface as the limit of this sum as the partition becomes finer (max rectangle size approaches zero).
  • πŸ“‰ The script provides an example of approximating the volume under the function f(x, y) = 9 - x^2 - y^2 over a specified rectangular region using four rectangles.
  • πŸ“ The example demonstrates choosing points at the bottom left corner of each rectangle for simplicity and calculating the approximate volume.
  • πŸ“Š The final step in the example is to evaluate the function at the chosen points, multiply by the area of the rectangles, and sum these products to get an approximation of the volume.
Q & A

  • What is the analogy between finding the area under a curve in single variable calculus and multivariable calculus?

    -In single variable calculus, the area under a curve is found by summing the areas of rectangles under the curve. Similarly, in multivariable calculus, the volume under a surface is found by summing the volumes of small boxes (or 'rectangles' in three dimensions) that approximate the surface.

  • How does increasing the number of subdivisions affect the approximation of the area under a curve or volume under a surface?

    -Increasing the number of subdivisions results in smaller rectangles or boxes, which leads to a better approximation of the area or volume. As the subdivisions become infinitesimally small, the approximation becomes more accurate, approaching the true area or volume.

  • What is the significance of choosing a specific point (Xk, Yk) within each rectangle in the multivariable calculus context?

    -The specific point (Xk, Yk) is chosen to determine the height of the box above the rectangle. This height is the value of the function at that point, which is then multiplied by the area of the base of the box to find the volume of the box.

  • Why does the choice of the point (Xk, Yk) within the rectangle not matter in the limit as the number of subdivisions approaches infinity?

    -In the limit, as the number of subdivisions approaches infinity and the size of each rectangle becomes infinitesimally small, the exact location of the point (Xk, Yk) within the rectangle has less impact on the overall approximation, because the error introduced by this choice becomes negligible.

  • What does the term 'partition' refer to in the context of approximating the volume under a surface?

    -A partition refers to the division of the domain into a finite number of smaller rectangles (or boxes). The quality of the approximation depends on the size of the largest rectangle in the partition; as this size approaches zero, the approximation improves.

  • How is the volume of each small box in the partition calculated?

    -The volume of each small box is calculated by multiplying the area of the base of the box (given by the product of Ξ”Xk and Ξ”Yk) by the height of the box, which is the function value F(Xk, Yk) at the chosen point within the box.

  • What is the mathematical notation used to represent the sum of the volumes of all small boxes in a partition?

    -The sum of the volumes of all small boxes is represented by the summation notation Ξ£, where the sum is taken from 1 up to n, representing the number of small boxes, and each term in the sum is the product of the function value at a point and the area of the base of the corresponding box.

  • What is the definition of the volume under a surface using Riemann integration?

    -The volume under a surface is defined as the limit of the sum of the volumes of the small boxes as the partition becomes finer (i.e., as the maximum size of the rectangles in the partition approaches zero), which corresponds to an increasing number of rectangles.

  • How is the function f(x, y) = 9 - x^2 - y^2 used as an example in the script to demonstrate the approximation of volume under a surface?

    -The function f(x, y) = 9 - x^2 - y^2 is evaluated at specific points within four rectangles that partition the domain. The volume under the surface is approximated by summing the volumes of these four rectangles, each with a height given by the function value at the bottom left corner of the rectangle.

  • What is the final step in the process of finding the exact volume under a surface using the method described in the script?

    -The final step is to take the limit of the sum of the volumes of the small boxes as the partition becomes finer, which means increasing the number of subdivisions until the rectangles are infinitesimally small, thus obtaining an exact value for the volume under the surface.

Outlines
00:00
πŸ“ Understanding the Area Under a Surface in Multivariable Calculus

This paragraph explains the concept of finding the area under a surface in multivariable calculus, which is analogous to finding the area under a curve in single-variable calculus. The idea is to divide the region into numerous small rectangles (in single-variable) or boxes (in multivariable), calculate the area or volume of each, and sum them up for an approximation. By increasing the number of these subdivisions, the approximation becomes more accurate. The paragraph details the process of breaking down the region, choosing points within each box, and how the approximation improves as subdivisions increase.

05:00
πŸ” Formal Definition of Volume under a Surface

This paragraph delves into the formal definition of volume under a surface using Riemann integration. It discusses partitioning the region into small rectangles, calculating the area of each, and choosing a specific point within each rectangle for the height. The volume is approximated by summing the volumes of these small boxes, with the height given by the function value at the chosen points. The paragraph introduces the concept of taking a limit where the size of the largest rectangle approaches zero and the number of rectangles approaches infinity, leading to an exact definition of volume under the surface.

Mindmap
Keywords
πŸ’‘Area Under the Curve
The term 'Area Under the Curve' refers to the space enclosed by a curve and the x-axis in a single-variable calculus problem. In the video, this concept is introduced as a foundational idea for understanding multivariable calculus problems. The script uses the analogy of breaking down the area into rectangles to approximate the total area under the curve, which is then extended to multivariable problems where 'boxes' are used instead.
πŸ’‘Multivariable Calculus
Multivariable Calculus is a branch of calculus that deals with functions of multiple variables, extending the concepts of single-variable calculus to higher dimensions. The video script discusses how the process of finding areas under curves in single-variable calculus is analogous to finding volumes in multivariable calculus, using the concept of partitioning the domain into small boxes.
πŸ’‘Rectangles and Boxes
In the context of the video, 'Rectangles and Boxes' are geometric shapes used to approximate areas and volumes under functions. For single-variable functions, rectangles are used to approximate the area under the curve, while for multivariable functions, three-dimensional 'boxes' are used to approximate volumes. The script explains how increasing the number of these shapes leads to a better approximation of the actual area or volume.
πŸ’‘Approximation
Approximation in the video refers to the method of estimating the area or volume under a function by breaking it down into smaller, more manageable shapes (rectangles or boxes). The script emphasizes that as the number of subdivisions increases, the approximation becomes more accurate, with errors decreasing as the shapes become smaller.
πŸ’‘Error
The 'Error' mentioned in the script is the discrepancy between the actual area or volume under a function and the approximation obtained by summing the areas or volumes of the rectangles or boxes. The script explains that this error diminishes as the number of subdivisions increases, leading to a more precise approximation.
πŸ’‘Partition
A 'Partition' in the script refers to the division of a region into smaller, non-overlapping subregions, such as rectangles or boxes, for the purpose of approximation. The video discusses how a finer partition, where the maximum size of any individual rectangle or box approaches zero, leads to a better approximation of the area or volume.
πŸ’‘Riemann Sum
The 'Riemann Sum' is a method used in calculus to approximate the definite integral of a function. In the video, it is extended to multivariable calculus, where the sum of the products of the function values at chosen points (XK, YK) and the areas of the rectangles or boxes (Delta XK, Delta YK) is used to approximate the volume under a surface.
πŸ’‘Limit
The 'Limit' in the script is a fundamental concept in calculus that describes the value that a function or sequence approaches as the input approaches some value. In the context of the video, the limit is used to define the exact volume under a surface as the number of subdivisions approaches infinity, making the rectangles or boxes infinitesimally small.
πŸ’‘Function Value
The 'Function Value' at a point (XK, YK) is the output of the function for that specific input. In the script, the function value is used to determine the height of the box in multivariable calculus, which is then multiplied by the base area to find the volume of the box.
πŸ’‘Volume
In the video, 'Volume' refers to the three-dimensional space enclosed under a surface, which is the main focus of the multivariable calculus problem discussed. The script explains how to approximate this volume by summing the volumes of small boxes and taking the limit as the size of these boxes approaches zero.
πŸ’‘Example
The 'Example' provided in the script illustrates the process of approximating the volume under a specific function within a given region. The function f(x, y) = 9 - x^2 - y^2 is used, and the region is partitioned into four rectangles. The script demonstrates how to calculate the approximate volume by summing the volumes of these rectangles, each with a height determined by the function value at a chosen point.
Highlights

Introduction to finding the area under a multivariable calculus problem analogous to a single variable problem.

Explanation of breaking the region into different rectangles to compute the area under a curve.

Comparison of multivariable calculus with single variable calculus using little boxes instead of rectangles.

Improving approximation by increasing the number of subdivisions into smaller rectangles or boxes.

Visual demonstration of one particular little box and the associated error in approximation.

Error in approximation decreases as the size of the subdivisions gets smaller.

Definition of the width (Delta X) and height (Delta Y) of the rectangles used in partitioning the domain.

Selecting specific points within the rectangles to plug into the function for height computation.

Point selection (XK, YK) within each rectangle and its impact on approximation.

Sum of the volumes of the little boxes provides an approximation for the volume under the surface.

Introduction to Riemann integration and defining volume using limits.

Taking the limit as the largest rectangle in the partition goes to zero area for a more accurate volume calculation.

Example using the function 9 - x^2 - y^2 over a rectangular region to approximate the volume.

Partitioning the region into four rectangles and choosing points at the bottom left corners for height calculation.

Final approximation of the volume by summing the volumes of the four rectangles and evaluating the function at chosen points.

Transcripts

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