Question

Find the volume of the set \(T\) bounded by the surfaces \(z=0, z=\sqrt{x^{2}+y^{2}},\) and \(x^{2}+y^{2}=4\).

Short answer

Answer: The volume of the set \(T\) is \(4\pi^2\) cubic units.

Step-by-step solution

Identify the boundaries and choose a coordinate system

First, let's visualize the given surfaces: 1. \(z=0\) is the xy-plane. 2. \(z=\sqrt{x^{2}+y^{2}}\) is a cone with the vertex at the origin. 3. \(x^{2}+y^{2}=4\) is a cylinder with radius 2 and centered at the origin. The given surfaces suggest that cylindrical coordinates might be useful for solving this problem; so let's switch to cylindrical coordinates \((r,\theta, z)\), where \(x= r\cos(\theta)\), \(y= r\sin(\theta)\), and \(z=z\).

Set up the limits for the triple integral

To find the volume of the set \(T\), we will integrate over the region \(T\). First, we'll find the limits of integration for each coordinate: 1. The variable \(r\) ranges from \(0\) to \(2\), because we are inside the cylinder with \(x^2+y^2=4\). 2. The variable \(\theta\) ranges from \(0\) to \(2\pi\) since we are considering the full revolution around the z-axis. 3. The variable \(z\) ranges from \(0\) (the xy-plane) to \(z=\sqrt{x^{2}+y^{2}}\) (the cone); since we are using cylindrical coordinates, we need to rewrite the boundary of the cone as \(z=\sqrt{r^2}\) or simply \(z=r\). Now we can set up the triple integral: $$V= \int\int\int_{T} r \, dz \, dr \, d\theta$$ With the specified limits of integration, this becomes: $$V= \int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{r} r \, dz \, dr \, d\theta$$

Evaluate the triple integral

Now we will evaluate the triple integral: $$V= \int_{0}^{2\pi} \int_{0}^{2} \left[\frac{1}{2}r^2z\right]_{0}^{r} dr \, d\theta = \int_{0}^{2\pi} \int_{0}^{2} \frac{1}{2}r^3 dr \, d\theta$$ Now, integrate with respect to \(r\): $$V= \int_{0}^{2\pi} \left[\frac{1}{8}r^4\right]_{0}^{2} d\theta = \int_{0}^{2\pi} \frac{1}{8}(16) d\theta = 2\pi$$ The next step is to integrate with respect to \(\theta\): $$V= \left[2\pi\theta\right]_{0}^{2\pi} = 2\pi(2\pi) - 2\pi(0) = 4\pi^2$$ So the volume of the set \(T\) is \(4\pi^2\) cubic units.

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates into three dimensions. It is particularly useful in situations where symmetry around a central axis is present, such as in the case of circular or cylindrical shapes. To convert from cylindrical to rectangular coordinates, these relationships are used:

• For the x-coordinate: \(x = r \times \text{cos}(\theta)\)
• For the y-coordinate: \(y = r \times \text{sin}(\theta)\)
• For the z-coordinate, it remains the same in both systems: \(z = z\)

By using cylindrical coordinates, integrating over volumes that have cylindrical symmetry becomes easier, as the coordinates align naturally with the shape of the object, simplifying the limits of integration and the integrand itself.

Triple Integral

A triple integral extends the idea of a double integral to three dimensions. It allows for the calculation of volumes, masses, and other properties of three-dimensional objects or regions. The triple integral is usually written as:
\[\int\int\int_T f(x, y, z) \text{d}V\]
Here, \(T\) represents the three-dimensional region of integration, while \(f(x, y, z)\) is the function being integrated, and \(dV\) denotes an infinitesimal volume element. In cylindrical coordinates, \(dV\) typically becomes \(r \text{d}z \text{d}r \text{d}\theta\), reflecting the geometry of the coordinate system.

Volume Calculation

The volume of a three-dimensional region can be calculated using a triple integral. Here's how the method typically works in a step-by-step manner:

• Identify the region of interest, its boundaries, and the most appropriate coordinate system.
• Express the limits for each integral based on the chosen coordinate system and the bounds of the region.
• Set up the integral with the function to be integrated, usually \(1\) for volume calculations, multiplied by the volume element of the chosen coordinate system.
• Perform the integration sequentially for each variable, evaluating the integral within the specified limits.

Using this method assures accurate representations of volume as it considers the shapes’ irregularities and varying cross-sections.

Real Analysis

Real analysis is the branch of mathematics that deals with the rigorous study of real numbers, sequences, series, continuity, differentiation, and integration. Within real analysis, the concept of volume using triple integrals is a vital application. It allows for a detailed understanding of the properties of three-dimensional space and provides the tools to calculate volumes beyond simple geometric shapes.

By employing methods from real analysis, such as integration techniques and limit theorems, we can precisely calculate the volume of complex regions mathematically. This is crucial in various fields, from physical sciences to engineering, wherever space and material properties are fundamental concerns.