Question

A region is space is described. Set up the triple integrals that find the volume of this region using rectangular, cylindrical and spherical coordinates, then comment on which of the three appears easiest to evaluate. The region enclosed by the cone \(z=\sqrt{x^{2}+y^{2}}\) and plane \(z=1 .\)

Short answer

Cylindrical coordinates appear easiest to evaluate, with simple and separable limits.

Step-by-step solution

Understand the Problem

We need to calculate the volume of the region bounded by the cone given by the equation \( z = \sqrt{x^2 + y^2} \) and the plane \( z = 1 \). This requires setting up triple integrals in rectangular (cartesian), cylindrical, and spherical coordinate systems.

Rectangular Coordinates Setup

To set up the triple integral in rectangular coordinates, observe the region where \( z \) runs between \( \sqrt{x^2 + y^2} \) (the cone) and 1 (the plane). The projection onto the \(xy\)-plane will be a circle of radius 1 centered at the origin. Thus, \[\int_{-1}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_{\sqrt{x^2+y^2}}^{1} \, dz \, dy \, dx. \]The limits on \(x\) and \(y\) describe a circular region.

Cylindrical Coordinates Setup

In cylindrical coordinates, \(x = r\cos\theta\), \(y = r\sin\theta\), and \(z = z\). The equation for the cone becomes \(z = r\) and the plane remains \(z = 1\). The limits are \(0 \leq r \leq 1\) (the radius of the circular cross-section), \(0 \leq \theta \leq 2\pi\), and \(r \leq z \leq 1\), giving the triple integral:\[\int_{0}^{2\pi} \int_{0}^{1} \int_{r}^{1} \, r \, dz \, dr \, d\theta.\]

Spherical Coordinates Setup

In spherical coordinates, \(x = \rho \sin\phi \cos\theta\), \(y = \rho \sin\phi \sin\theta\), and \(z = \rho \cos\phi\). The cone is defined by \(\phi = \frac{\pi}{4}\) and the plane at \(z=1\) corresponds to \(\rho\cos\phi = 1\). Therefore, \\[\int_{0}^{2\pi} \int_{0}^{\frac{\pi}{4}} \int_{\frac{1}{\cos\phi}}^{\sqrt{2}} \, \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta.\]The limits on \(\rho\) are determined by setting \(\rho\cos\phi = 1\), and \(\phi\) is limited by the cone's angle.

Analyze Easiest Integral to Evaluate

Evaluate the setup for each coordinate system. In cylindrical coordinates, the integrals over \(r\), \(\theta\), and \(z\) are easily separable and straightforward to compute due to limits being constants or simple functions. In both rectangular and spherical coordinates, the presence of square roots and trigonometric functions complicates the integration process.

Key concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are one of the most fundamental coordinate systems. They represent points in space using three values: \(x\), \(y\), and \(z\). These coordinates make it easier to represent geometric problems in terms of lines, planes, and other standard shapes. However, when calculating volumes involving non-standard shapes like cones or spheres, rectangular coordinates can become cumbersome due to complex integration limits.
To find the volume of a region described by both a cone and a plane using rectangular coordinates, we first establish the integration boundaries. The given cone equation is \(z = \sqrt{x^2 + y^2}\) and extends until the plane \(z = 1\). As a result, the bounds for integration are:

• \(-1 \leq x \leq 1\)
• \(-\sqrt{1-x^2} \leq y \leq \sqrt{1-x^2}\)
• \(\sqrt{x^2 + y^2} \leq z \leq 1\)

These boundaries result in a triple integral to describe the volume, although the square root in the limits can make it complicated to evaluate compared to other coordinate systems.

Cylindrical Coordinates

Cylindrical coordinates offer a great alternative when dealing with problems involving circular shapes. These coordinates convert a point in space into \(r\) (radius), \(\theta\) (angle in the \(xy\)-plane), and \(z\) (height). This system is particularly useful for geometries that are symmetrical around an axis, like cylinders and cones.
For this exercise, the equations \(x = r\cos\theta\) and \(y = r\sin\theta\) transform our problem to fit cylindrical coordinates, simplifying the process of setting up an integral. The cone equation becomes \(z = r\), and the plane remains \(z = 1\). Thus, the boundaries for integration are:

• \(0 \leq r \leq 1\)
• \(0 \leq \theta \leq 2\pi\)
• \(r \leq z \leq 1\)

These simpler integration limits, along with the straightforward multiplication by \(r\) during integration, often make the cylindrical coordinate system one of the easiest to evaluate when dealing with volumes of these symmetrical regions.

Spherical Coordinates

Spherical coordinates are best suited for dealing with spheres and other shapes extending outwards in all directions. These consist of \(\rho\) (the distance from the origin), \(\phi\) (the angle from the positive \(z\)-axis), and \(\theta\) (angle in the \(xy\)-plane). For the problem at hand, this setup simplifies dealing with the spatial region defined by the cone and plane.
In these coordinates, the cone's condition is expressed by \(\phi = \frac{\pi}{4}\), meaning the angle from the \(z\)-axis remains constant. The plane condition \(z=1\) translates to \(\rho \cos\phi = 1\). The bounds become:

• \(0 \leq \theta \leq 2\pi\)
• \(0 \leq \phi \leq \frac{\pi}{4}\)
• \(\frac{1}{\cos\phi} \leq \rho \leq \sqrt{2}\)

Even though spherical coordinates offer a straightforward geometric perspective, the integration can become tricky due to the involved trigonometric functions and variables like \(\sin\phi\) in the limits.

Volume Calculation

Volume calculation through triple integrals involves integrating over a solid region defined by the limits of the chosen coordinate system. The volume is represented by the triple integral, which sums up the infinitesimal contributions over every point within the region. Depending on the symmetry and complexity of the region, different coordinate systems present different levels of difficulty when setting up and evaluating these integrals.
The volume under consideration here is constrained by a conical surface and a plane. Each coordinate system offers a method to set up these bounds, but the symmetry around one or more axes often guides the choice towards cylindrical coordinates, which has more straightforward constant limits, leading to an easier calculation and reduction in complexity while ensuring all portions of the region are included.

Coordinate Transformations

Coordinate transformations allow us to switch from one system to another, making complex problems easier or conforming to a specific symmetry in the region or problem. By transforming coordinates, we're able to exploit natural geometrical traits in problems, such as spherical symmetry or circular patterns.
When facing complex regions like those in this exercise, identifying the best coordinate system significantly simplifies the work involved. Each transformation makes different aspects of the integral easier to visualize and compute:

• Transforming to cylindrical coordinates can simplify circular base problems, reducing one horizontal plane dimension to a radius.
• Spherical coordinates simplify volumes in radial symmetry cases and can reduce complex volume calculations involving multiple dimensions.

Understanding when and how to use these transformations is crucial, not only to simplify calculations but also to ensure a clear spatial understanding of the problem.