Question

Write a triple integral in cylindrical coordinates for the volume inside the cylinder \(x^{2}+y^{2}=4\) and between \(z=2 x^{2}+y^{2}\) and the \((x, y)\) plane. Evaluate the integral.

Short answer

The volume is \(16\pi\).

Step-by-step solution

- Convert to Cylindrical Coordinates

Recognize that in cylindrical coordinates, the equations transform as follows: - For the cylinder: \(x^2 + y^2 = r^2 = 4\), so \(r = 2\).- For the plane: \(z = 0\).- For the surface: \(z = 2x^2 + y^2\). Converting this to cylindrical coordinates gives \(z = 2r^2\).

- Determine the Bounds

The bounds for \(r\) will be from 0 to 2 because it is confined within the cylinder. The bounds for \(\theta\) will range from 0 to \(2\pi\). The bounds for \(z\) will range from \(0\) to \(2r^2\). Hence, \[0 \leq r \leq 2, \quad 0 \leq \theta \leq 2\pi, \quad 0 \leq z \leq 2r^2\]

- Write the Triple Integral

In cylindrical coordinates, the triple integral for the volume \(V\) becomes \[ V = \int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{2r^2} r \, dz \, dr \, d\theta \] Note the additional \(r\) in the integrand, which accounts for the Jacobian determinant when converting from Cartesian to cylindrical coordinates.

- Evaluate the Integral

First, the innermost integral with respect to \(z\) is: \[ \int_{0}^{2r^2} r \, dz = r[z]_{0}^{2r^2} = r(2r^2 - 0) = 2r^3 \] Next, integrating with respect to \(r\): \[ \int_{0}^{2} 2r^3 \, dr = 2\left[ \frac{r^4}{4} \right]_{0}^{2} = 2\left( \frac{16}{4} - 0 \right) = 2 \cdot 4 = 8 \] Finally, integrating with respect to \(\theta\): \[ \int_{0}^{2\pi} 8 \, d\theta = 8[\theta]_{0}^{2\pi} = 8(2\pi - 0) = 16\pi \] Hence, the volume is \[ V = 16\pi \]

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height component. Instead of representing points with \(x\), \(y\) and \(z\) like in Cartesian coordinates, we use \(r\), \(\theta\), and \(z\). Here, \(r\) is the distance from the origin to the projection of the point onto the \(xy\)-plane, \(\theta\) is the angle formed with the positive \(x\)-axis, and \(z\) is the same as in Cartesian coordinates.
The transformation equations between Cartesian and cylindrical coordinates are:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)
• \(r^2 = x^2 + y^2\)
• \(\theta = \tan^{-1}(\frac{y}{x})\)

This system simplifies calculations, especially for problems involving symmetry around an axis like cylinders and spherical shapes.

Triple Integral

A triple integral allows us to compute the volume of a three-dimensional region by integrating a function over a solid in space. When performed in cylindrical coordinates, the triple integral becomes more manageable for problems with cylindrical symmetry. In cylindrical coordinates, the volume element, or the differential volume, changes from \(dV = dx \, dy \, dz\) in Cartesian coordinates to \(dV = r \, dr \, d\theta \, dz\). The extra factor of \(r\) accounts for the Jacobian determinant, which ensures the volume element correctly scales from Cartesian to cylindrical coordinates.
To set up a triple integral in cylindrical coordinates, follow these steps:

• Identify the bounds for \(r\), \(\theta\), and \(z\).
• Write the integrand including the \(r\) factor.
• Integrate in the order indicated by the integral bounds.

Volume Calculation

Volume calculation using triple integrals involves breaking the solid into infinitesimally small volume elements and summing them. In our problem, the volume inside the cylinder \(x^2 + y^2 = 4\) and between the surfaces \(z = 2x^2 + y^2\) and \(z = 0\) can be calculated as follows:

• Convert the bounds to cylindrical coordinates: \(0 \leq r \leq 2\), \(0 \leq \theta \leq 2\pi\), \(0 \leq z \leq 2r^2\).
• Set up the triple integral: \[ V = \int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{2r^2} r \, dz \, dr \, d\theta \]
• Integrate step by step, starting with the innermost integral.

By systematically integrating from the innermost to the outermost bounds, we obtain the final volume \(V = 16\pi\).

Change of Variables

Change of variables is a method used to simplify integrals by transforming them into a more convenient coordinate system. In this exercise, we switch from Cartesian coordinates to cylindrical coordinates to take advantage of the problem's symmetry.
Changing the variables involves:

• Identifying the appropriate transformation equations (e.g., \(x = r\cos\theta \)).
• Rewriting the region's boundaries in the new coordinates.
• Adjusting the volume element to include the Jacobian determinant (here, \(r \) in cylindrical coordinates).

This process reduces the complexity of the integrals and makes them easier to evaluate. Such transformations are essential in multi-variable calculus and have wide applications in physics and engineering.