Problem 100 - Calculus Early Transcendentals

Question

Compute the volume of the following solids. The wedge sliced from the cylinder \(x^{2}+y^{2}=1\) by the planes \(z=1-x\) and \(z=x-1\)

Step-by-Step Solution

Answer

Question: Compute the volume of the wedge created by the intersection of a cylinder defined by the equation \(x^2 + y^2 = 1\), and two planes: \(z = 1 - x\) and \(z = x - 1\). Answer: The volume of the wedge can be calculated by evaluating the triple integral, set up in cylindrical coordinates: \(V = \left|\int_{0}^{1} [2r^2\sin(2\pi - 2\arccos{(1/r)}) - 2r^2\sin(0)] \,dr\right|\).

Step 1: Set up the problem in cylindrical coordinates

Since we are working with a cylinder, it's best to use cylindrical coordinates. Replace \(x^2 + y^2\) by \(r^2\), where \(r\) is the radial distance from the \(z\)-axis. Our equation becomes: \(r^2 = 1\) The two planes can also be written in cylindrical coordinates. First, solve the plane equations for \(z\): \(z = 1 - x\) \(z = x - 1\) Now, replace \(x\) by \(r\cos\theta\): \(z = 1 - r\cos\theta\) \(z = r\cos\theta - 1\)

Step 2: Find the boundary of the intersection

We need to figure out the boundaries of the intersection between the cylinder and two planes. To do this, first find the intersection of the planes by setting the \(z\)-coordinates equal to each other: \(1 - r\cos\theta = r\cos\theta - 1\) Now solve for \(r\cos\theta\): \(r\cos\theta = 1\)

Step 3: Set up the triple integral

Our goal is to compute the volume by integrating over the region with the appropriate kernel. In cylindrical coordinates, we have: \(dV = r\,dr\,d\theta\,dz\) The bounds for \(r\) are \(0 \leq r \leq 1\), for \(\theta\), we need to determine the bounds where the intersection of the planes occurs with the cylinder: \(0 \leq \theta \leq 2\pi - 2\arccos{(1/r)}\) And for \(z\), \(1-r\cos\theta \leq z \leq r\cos\theta - 1\): \(\int_{0}^{1}\int_{0}^{2\pi - 2\arccos{(1/r)}}\int_{1-r\cos\theta}^{r\cos\theta - 1} r\,dz\,d\theta\,dr\)

Step 4: Perform the integration

Let's first integrate with respect to \(z\): \(\int_{0}^{1}\int_{0}^{2\pi - 2\arccos{(1/r)}} r[(r\cos\theta - 1) - (1 - r\cos\theta)]\,d\theta\,dr\) This simplifies to: \(\int_{0}^{1}\int_{0}^{2\pi - 2\arccos{(1/r)}} 2r^2\cos\theta\,d\theta\,dr\) Next, integrate with respect to \(\theta\): \(\int_{0}^{1} [2r^2\sin\theta]_{0}^{2\pi - 2\arccos{(1/r)}} \,dr\) Finally, integrate with respect to \(r\), remembering to add the absolute value because the volume is always positive: \(V = \left|\int_{0}^{1} [2r^2\sin(2\pi - 2\arccos{(1/r)}) - 2r^2\sin(0)] \,dr\right|\) Evaluate this integral to get the volume of the wedge.

Key Concepts

Cylindrical Coordinates

When tackling problems related to cylinders, switching to cylindrical coordinates can significantly simplify the math.
In cylindrical coordinates, a point in space is represented by three values:

• \(r\) - the radial distance from the origin (akin to the radius in polar coordinates).
• \(\theta\) - the angle of rotation from the positive x-axis.
• \(z\) - the height above the xy-plane, which is a natural extension from the Cartesian z-coordinate.

For the specific problem at hand, we translate the cylinder equation \(x^2 + y^2 = 1\) to \(r^2 = 1\).
This equation states that the radial distance is constant at 1, defining a unit cylinder.

Triple Integral

Computing the volume of a solid shape often requires the application of a triple integral.
This involves integrating a function over a three-dimensional region. In this case, the function to integrate is simply the differential volume \(dV = r \, dr \, d\theta \, dz\).

• The order of integration can be carefully selected to simplify the process. Here, the integration occurs first in \(z\), then \(\theta\), and finally in \(r\).
• Each integral has specific bounds, derived from the geometry formed by the cylinder and the intersecting planes.

This structured integration efficiently accumulates the total volume of the defined solid within the given boundaries.

Solid Geometry

Understanding the geometry of combined solids is crucial.
In this case, the wedge is defined by intersecting a unit cylinder with two planes.
By examining the equations \(z = 1 - x\) and \(z = x - 1\) in cylindrical coordinates:

• These describe planes slicing through the cylinder, producing a wedge-shaped section.
• The intersection conditions determine limits of integration and essential properties like symmetry.

Evaluating the shape in cylindrical terms allows insights into how these geometric elements interact, crucial for effectively setting up the integral bounds.