Problem 80 - Thomas Calculus

Question

Cylinder and planes Find the volume of the region enclosed by the cylinder \(x^{2}+y^{2}=4\) and the planes \(z=0\) and \(x+y+z=4 .\)

Step-by-Step Solution

Answer

The volume is \(\frac{16}{3} \times 2\pi\).

Step 1: Understand the Problem

The problem is to find the volume enclosed by a cylindrical surface described by the equation \(x^2 + y^2 = 4\) and two planes, \(z = 0\) and \(x + y + z = 4\). This involves three-dimensional geometry.

Step 2: Visualization

Visualize the cylinder which extends infinitely along the \(z\)-axis bounded at the bottom by the plane \(z = 0\), and at the top by the plane \(x + y + z = 4\). The cylinder is centered at the origin with a radius of 2.

Step 3: Equation of the Bounds in 3D Space

Recognize that the upper boundary is given by \(z = 4 - x - y\) for the plane. Thus, the volume is defined between \(z = 0\) to \(z = 4 - x - y\).

Step 4: Set up the Volume Integral

Since the problem involves rotational symmetry about the \(z\)-axis, switch to polar coordinates: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). The corresponding cylinder equation becomes \(r^2 = 4\), therefore \(r = 2\). Set up the triple integral for volume using the limits for \(r\) from \(0\) to \(2\), \(\theta\) from \(0\) to \(2\pi\), and \(z\) from \(0\) to \(4 - r\cos(\theta) - r\sin(\theta)\).

Step 5: Evaluate the Integral

The volume \(V\) is given by:\[ V = \int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{4 - r(\cos\theta + \sin\theta)} r \, dz \, dr \, d\theta \]Evaluate this integral by first integrating with respect to \(z\), then \(r\), and finally \(\theta\) to find the enclosed volume.

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are particularly useful for solving problems involving cylindrical symmetry. The system uses three parameters: \(r\), \(\theta\), and \(z\). Here, \(r\) represents the radial distance from the \(z\)-axis, \(\theta\) is the angle measured from the positive \(x\)-axis within the plane parallel to \(xy\), and \(z\) is the height axis. In our exercise, the cylindrical coordinate transformation simplifies the problem by utilizing the symmetry of the cylindrical region.To transform from Cartesian coordinates (\(x, y, z\)) to cylindrical:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)
• \(z = z\)

For the cylinder with equation \(x^2 + y^2 = 4\), we find that \(r^2 = 4\), which simplifies to \(r = 2\), indicating the radius. Using these coordinates allows us to use triple integration effectively.

Triple Integration

Triple integration is a method used to calculate the volume of a three-dimensional region. It extends the concepts of single and double integrals to three dimensions. This is performed by integrating a function over a three-dimensional space. In solving the exercise, triple integration is necessary because the volume being calculated lies within a defined region across three parameters: \(r\), \(\theta\), and \(z\).The order of integration in our problem is:

• First, with respect to \(z\) (from the bottom plane \(z = 0\) to the upper boundary \(z = 4 - r\cos(\theta) - r\sin(\theta)\))
• Then, with respect to \(r\) (spanning from 0 to the outer radius 2 of the cylindrical base)
• Lastly, with respect to \(\theta\) (as it sweeps a full circle from \(0\) to \(2\pi\))

This method effectively slices the volume into smaller pieces, calculates the volume for each piece, and sums them up.

3D Geometry

Three-dimensional geometry explores the spatial relationships and dimensions of objects in three spaces. When working with 3D geometry, we often need to consider how shapes interact across different planes and axes.In the exercise, we observe a cylindrical region bounded by a basic cylinder equation \(x^2 + y^2 = 4\), which outlines a circular base of radius 2. This cylinder is bounded vertically by two planes: at the bottom by \(z = 0\), and at the top by the inclined plane \(x+y+z=4\). Understanding these boundaries is crucial.Each point within these bounds represents a valid point within the defined volume. The use of 3D geometry helps in visualizing such regions and understanding how integrals are used to calculate the total volume.

Volume Calculation

Volume calculation in the context of mathematical integration involves finding the total space enclosed within a three-dimensional region. For cylindrical regions bounded by planes, triple integration in cylindrical coordinates becomes an efficient approach.The given exercise involves integrating a function over a specified region: \[ V = \int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{4 - r(\cos\theta + \sin\theta)} r \, dz \, dr \, d\theta \]Breaking it down:

• \(z\) integration is performed first, limited by the planes \(z=0\) to \(z=4 - r(\cos\theta + \sin\theta)\).
• \(r\) and \(\theta\) integration controls the radius and circumference, spanning from 0 to 2 and 0 to \(2\pi\), respectively.

The entire expression accounts for each infinitesimal sector of the volume within the cylindrical bounds. Learning how to set limits and choose the right coordinate system is essential for easy volume calculations in such scenarios.