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 cylinder \(x^{2}+y^{2}=1\) and planes \(z=0\) and \(z=1\).

Short answer

Polar and cylindrical coordinates are the easiest to evaluate for this region due to their natural alignment with cylindrical symmetry.

Step-by-step solution

Rectangular Coordinates Setup

In rectangular coordinates, the region is a cylinder with the equation \(x^{2} + y^{2} = 1\) and it's bounded between the planes \(z = 0\) and \(z = 1\). The limits for \(x\) and \(y\) are the circle defined by \(x^{2} + y^{2} \leq 1\). The triple integral to find the volume in rectangular coordinates is:\[\int_{0}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \int_{0}^{1} dz \, dy \, dx.\]

Evaluation of Rectangular Integral

Evaluate the innermost integral with respect to \(z\). It evaluates simply to 1, because the integral is from 0 to 1:\[\int_{0}^{1} dz = 1.\]This simplifies the triple integral to:\[\int_{0}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \, dy \, dx.\]

Polar Coordinates Setup

Convert the region to polar coordinates where \(x=r\cos(\theta)\), \(y=r\sin(\theta)\), and \(r^2 = x^2 + y^2 \leq 1\). The limits for \(r\) are from 0 to 1 and for \(\theta\) from 0 to \(2\pi\), with \(z\) ranging from 0 to 1. The volume in polar coordinates becomes:\[\int_{0}^{2\pi} \int_{0}^{1} \int_{0}^{1} r \, dz \, dr \, d\theta.\]

Evaluation of Polar Integral

Perform the integration starting with \(z\):\[\int_{0}^{1} dz = 1.\]Next, integrate with respect to \(r\):\[\int_{0}^{1} r \, dr = \frac{1}{2}.\]Then, integrate with respect to \(\theta\):\[\int_{0}^{2\pi} d\theta = 2\pi.\]The product of these results gives the volume: \(2\pi \times \frac{1}{2} = \pi.\)

Cylindrical Coordinates Setup

The region in cylindrical coordinates is similarly defined since it naturally fits cylindrical symmetry. Use the limits already identified: \(r\) from 0 to 1, \(\theta\) from 0 to \(2\pi\), and \(z\) from 0 to 1. The integral is:\[\int_{0}^{2\pi} \int_{0}^{1} \int_{0}^{1} r \, dz \, dr \, d\theta.\]

Evaluation of Cylindrical Integral

Since the setup is the same as polar coordinates, the evaluation steps are identical:1. Integrate over \(z\): gives 1.2. Integrate over \(r\): gives \(\frac{1}{2}\).3. Integrate over \(\theta\): gives \(2\pi\).Multiply the results to find the volume: \(2\pi \times \frac{1}{2} = \pi.\)

Spherical Coordinates Attempt

The cylinder does not align well with spherical coordinates because they are optimal for spherical symmetry, not cylindrical. Attempting this isn't straightforward without complicating the limits and transformation equations, and thus, is inefficient for this shape.

Evaluation Comparison and Conclusion

After reviewing all methods, both polar and cylindrical coordinates offer the simplest form due to natural alignment with the region's symmetry and requiring fewer transformations or adjustments. Rectangular was straightforward initially but required dealing with square roots in the limits. Spherical was least efficient given its complexity for this shape.

Key concepts

Rectangular Coordinates

Rectangular coordinates are the most basic and familiar form of coordinates in mathematics. They involve the three axes: x, y, and z, forming a three-dimensional coordinate system. Each point in space is defined by a tuple \((x, y, z)\).

When dealing with geometric shapes like cylinders or cubes, rectangular coordinates provide a straightforward way to express volumes and regions in space. In this exercise, the region has been enclosed by the cylinder \((x^2 + y^2 = 1)\) and planes \(z = 0\) and \(z = 1\).

This region is suited for rectangular coordinates because we directly apply the limits associated with each dimension. Here, x and y are defined within the circle \((x^2 + y^2 \leq 1)\) on the xy-plane, while z is bounded by 0 and 1. Though setting up the integral was intuitive, complications arise due to circular boundaries that introduce square root expressions in the y-limits. This makes actual calculation more complex.

Cylindrical Coordinates

Cylindrical coordinates are particularly effective for regions symmetrical around an axis, such as cylinders. They use three parameters: radius r, angle \(\theta\), and height z.

This system is a natural match for cylindrical shapes because of its direct alignment with their symmetry. Here, the radius \(r\) ranges from 0 to 1, angle \(\theta\) wraps fully around from 0 to \(2\pi\), and z extends between the bounds of the planes at 0 and 1.

In this exercise, transforming the region to cylindrical coordinates simplifies calculations as it takes advantage of this symmetry. The integrals set up in cylindrical coordinates are identical to those in polar coordinates, but the cylindrical form is naturally more intuitive for dealing with vertical structures due to the addition of the z-axis. This eliminated square roots, simplifying the volume integration considerably.

Spherical Coordinates

Spherical coordinates are not ideal for the current exercise where a cylinder is involved, as they are tailored most effectively for regions with spherical symmetry, such as spheres or hemispheres.

A point in spherical coordinates is determined by three parameters: the radial distance \(\rho\), the polar angle \(\phi\) (with respect to the z-axis), and the azimuthal angle \(\theta\) (around the z-axis).

The complication here arises from transforming cylindrical regions into spherical coordinates, which do not easily share the same symmetries. Attempting to set limits for a cylinder in spherical form complicates them unnecessarily and adds complexity rather than simplification. Therefore, while spherical coordinates are a powerful tool for certain regions, they are inefficient for this particular exercise.

Volume Calculation

Calculating volume through triple integrals involves summing up infinitely small volumes within a defined region. This is achieved by dividing the region into infinitesimally thin slices or elements, based on the coordinate system.

With rectangular coordinates, these elements are small rectangular prisms. For cylindrical coordinates, the elements are cylindrical shells, and for spherical coordinates, they are spherical shells.

This exercise illustrates how volume can be computed by integrating over three independent variables representing these smaller volumes. While the ultimate goal remains the same across each coordinate system—finding the total volume—the method and simplicity of finding that result can vary greatly. By integrating through different coordinate transformations, we are often able to simplify the mathematics involved, particularly when the shape aligns neatly with the chosen coordinate system.

Coordinate Transformation

A coordinate transformation involves switching from one coordinate system to another to leverage certain symmetries or simplify the mathematical process. It can turn a complex integral in one system into a simpler form in another system.

In this exercise, converting the given problem into cylindrical coordinates optimizes the setup and computation time due to the symmetry of the cylinder. By aligning the problem with a coordinate system that matches its shape naturally, transformations reduce the mathematical complexity, often simplifying limits of integration and the functions within the integral.

This highlights why learning to choose the proper coordinate system for integration is crucial. Even though transformations require initial setup in terms of conversion equations, the simplicity gained through strategic transformations often results in faster and cleaner solutions.