Double integrals 3 | Double and triple integrals | Multivariable Calculus | Khan Academy

Khan Academy
13 Aug 200808:03
EducationalLearning
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TLDRIn this mathematical exploration, the video delves into the process of calculating the volume between the surface defined by xy squared and the xy-plane within specified limits for x and y. Initially, integration with respect to x was performed, resulting in a volume of 2/3. The video then demonstrates an alternative approach by integrating with respect to y first, holding x constant, and confirming the same result. This methodological validation reinforces the concept of double integration in determining volumes and showcases the power of mathematical consistency.

Takeaways
  • 📈 The problem involves calculating the volume between the surface defined by the equation z = xy^2 and the xy-plane over the region 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1.
  • 🔄 The method used in the previous video was integrating with respect to x first, then y, but this time we will reverse the order of integration to verify the result.
  • 🎨 The visualization of the problem involves a 3D graph with the x-axis, y-axis, and z-axis, where the xy-plane is at the bottom and the surface lies above it.
  • 🖋️ The first step in the new method is to hold x constant and calculate the area under the curve for a given x value, treating x as a constant.
  • 🧩 The area under the curve for a fixed x is found by integrating dy * (xy^2) from y=0 to y=1, which gives us the area as a function of x.
  • 📚 The antiderivative of y^2 with respect to y is (y^3)/3, and when evaluated from y=0 to y=1, it results in the function (x/3) for a given x.
  • 🔢 To find the volume, the area function (x/3) is multiplied by dx, and then integrated with respect to x from 0 to 2.
  • 🌀 The antiderivative of x with respect to x is (x^2)/2, and when evaluated from x=0 to x=2, it results in the volume formula (x^2)/6.
  • 📈 The final volume calculated by integrating with respect to y first and then x is 4/6, which simplifies to 2/3, matching the result from the previous method.
  • 🔄 This demonstrates the principle of mathematical consistency, where the same result can be obtained by different合法的 integration orders.
  • 🎓 The video emphasizes the importance of understanding the order of integration and the ability to visualize mathematical problems in three dimensions.
Q & A

  • What was the main topic of the video?

    -The main topic of the video was calculating the volume between the surface defined by the equation z = xy^2 and the xy-plane over the region where x ranges from 0 to 2 and y ranges from 0 to 1.

  • How did the video demonstrate the concept of integrating with respect to x first?

    -The video demonstrated the concept by first fixing a value of y, then calculating the area under the curve for that y, and finally integrating with respect to x to find the volume.

  • What was the initial method used to calculate the volume?

    -The initial method used to calculate the volume was by integrating with respect to x first and then integrating with respect to y.

  • Why was it important to try integrating in a different order?

    -Integrating in a different order was important to verify the correctness of the result, showing that the volume calculated would be the same regardless of the order of integration.

  • How did the video explain the process of integrating with respect to y first?

    -The video explained the process by holding x constant, treating it as a constant, and then calculating the area under the curve for a given x. This area was then multiplied by dx to account for the depth, and integrated with respect to x to find the volume.

  • What was the antiderivative of y^2 when integrating with respect to y first?

    -When integrating with respect to y first, the antiderivative of y^2 was y^3/3, considering x as a constant.

  • How did the video represent the change in area depending on the value of x?

    -The video represented the change in area depending on the value of x by showing that the area is x/3 for any given x, and this area changes as x varies from 0 to 2.

  • What was the final result of the volume calculation?

    -The final result of the volume calculation was 2/3, which was the same as when integrating with respect to x first and then y.

  • How did the video conclude the demonstration?

    -The video concluded by showing that the volume under the surface z = xy^2 and the xy-plane is 2/3, confirming that the result was consistent regardless of the order of integration.

  • What was the significance of the graph and surface visualization in the video?

    -The graph and surface visualization in the video helped to provide an intuitive understanding of the volume being calculated and how the integration process relates to the shape of the surface.

  • What was the antiderivative of x with respect to the final volume calculation?

    -In the final volume calculation, the antiderivative of x was x^2/2, which was used to evaluate the area of the green surface depending on the value of x.

Outlines
00:00
📚 Double Integral for Volume Calculation

This paragraph discusses the process of calculating the volume between a surface defined by the equation xy squared and the xy-plane with given constraints on the x and y axes. Initially, the speaker explains the method previously used, which involved integrating with respect to x first and then y, resulting in the answer 2/3. The speaker then introduces an alternative approach by integrating with respect to y first, to verify the same result. The explanation includes a detailed description of the graph and the steps to calculate the area under the curve for a fixed value of x, followed by integrating with respect to x to find the volume. The paragraph concludes with a reiteration of the graph and the calculated volume, emphasizing the importance of understanding the order of integration in double integrals.

05:02
📈 Evaluating Inner and Outer Integrals for Volume

The second paragraph delves into the specifics of evaluating the inner integral with respect to y, holding x constant, and then using the result to calculate the total volume. The speaker clarifies that the area represented in green is a function of x and depends on the chosen value of x. The process of evaluating the inner integral is explained, leading to the antiderivative of y squared, which is y cubed over 3 when x is constant. The outer integral with respect to x is then discussed, with the antiderivative of x being x squared over 2. The final step involves evaluating this expression at the bounds of integration (from 0 to 2) and multiplying by 1/3 to get the volume, which is confirmed to be 2/3, consistent with the previous method. The paragraph ends with a reflection on the successful completion of the volume calculation and a visual appreciation of the graph.

Mindmap
Keywords
💡Volume
In the context of the video, 'volume' refers to the three-dimensional space occupied by the region between the surface defined by the equation xy^2 and the xy-plane, bounded by x ranging from 0 to 2 and y from 0 to 1. The calculation of volume is central to the video's theme, demonstrating the process of integrating with respect to x first and then y, and vice versa, to arrive at the same result of 2/3.
💡Integration
Integration is a fundamental concept in calculus that involves finding the antiderivative of a function and evaluating it over a given interval to compute areas or volumes. In the video, integration is used to calculate the area under a curve (for a fixed value of x) and then the volume under a surface by integrating over the given domain of x and y.
💡xy-plane
The xy-plane is the two-dimensional coordinate plane in a three-dimensional Cartesian coordinate system, where the axes represent the x and y dimensions. In the video, the xy-plane is used as a reference to define the region above which the surface (xy^2) exists, and thus the volume to be calculated is the space between this plane and the surface.
💡Surface
A 'surface' in the context of the video refers to the three-dimensional shape that is defined by the equation xy^2 in the Cartesian coordinate system. The surface is the geometric figure that is bounded below by the xy-plane and extends into the third dimension (z) based on the given equation.
💡Antiderivative
An antiderivative, also known as an indefinite integral, is a function that, when differentiated, yields the original function. In the video, finding the antiderivative is a crucial step in the process of integration, which is used to calculate the area under a curve and subsequently the volume under a surface.
💡Limits of Integration
The 'limits of integration' refer to the boundaries of the interval over which the integration is performed. In the video, the limits of integration are the values of x and y that define the region where the volume is being calculated, specifically x from 0 to 2 and y from 0 to 1.
💡Curve
In the context of the video, a 'curve' is the two-dimensional graph of a function, such as z = xy^2. The curve represents the intersection of the surface with planes parallel to the xy-plane at different values of x.
💡Area
The 'area' is a measure of the extent of a two-dimensional shape or surface. In the video, the area is a precursor to calculating the volume, as the volume is found by calculating the area under the curve for a fixed x and then integrating over the range of x.
💡Depth
In the context of the video, 'depth' refers to the extent in the third dimension (z-direction) of the region being studied. The depth is represented by the variable dx when integrating with respect to x to find the volume under the surface.
💡Graph
A 'graph' is a visual representation of the data or functions, typically consisting of points plotted on a coordinate system. In the video, the graph is used to visualize the curve defined by the function z = xy^2 and the volume under this surface in relation to the xy-plane.
💡Coordinate System
A 'coordinate system' is a grid used to specify points in a plane or space with a set of numbers. In the video, a three-dimensional Cartesian coordinate system is used, with axes representing the x, y, and z dimensions, to define the surface and calculate the volume.
Highlights

The video begins with a recap of a previous problem involving the volume between a surface and the xy-plane, specifically when x ranges from 0 to 2 and y from 0 to 1.

The method previously used involved integrating with respect to x first and then with respect to y, resulting in a volume of 2/3.

The video aims to confirm the correctness of the previous result by integrating in the opposite order, starting with respect to y.

The graph of the function is redrawn to provide visual context, emphasizing the x-axis, y-axis, and z-axis.

The xy-plane is defined with y ranging from 0 to 1 and x from 0 to 2, and the surface graph is approximated.

The focus is on the volume underneath the graph, which is the area under the curve integrated over the range of x.

A new method is introduced by holding x constant and treating it as a function of y squared, simplifying the area calculation.

The area under the curve for a given x is calculated by multiplying xy squared by dy and integrating from y=0 to y=1.

The volume under the entire surface is found by multiplying the area by dx, effectively adding depth to the 2D area.

The inner integral with respect to y is evaluated, resulting in a function of x, which is x/3.

The outer integral with respect to x is then calculated, with the antiderivative of x being x squared over 2.

The final volume calculation is performed by evaluating the x integral from 0 to 2, resulting in 4/6 or 2/3.

The video concludes by confirming that integrating in either order yields the same result, demonstrating the consistency of the mathematical approach.

The video ends with a visual representation of the rotated graph, reinforcing the understanding of the volume between the surface and the xy-plane.

The video serves as a practical example of applying integration techniques to find volumes, showcasing the importance of understanding different approaches.

Transcripts

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