Question

Find the following average values. The average of the squared distance between the origin and points in the solid cylinder \(D=\left\\{(x, y, z): x^{2}+y^{2} \leq 4,0 \leq z \leq 2\right\\}\)

Short answer

Answer: The average squared distance between the origin and points in the solid cylinder is 4.

Step-by-step solution

Define integral bounds

Define the integral bounds to calculate the triple integral, given the function \(f(x, y, z) = x^2 + y^2\) and the cylinder \(D=\{(x, y, z): x^{2}+y^{2} \leq 4,0 \leq z \leq 2\}\). The integration bounds for \(z\) are already given: \(0 \leq z \leq 2\). For the radius (x and y bounds), we can use the polar coordinates as \(0\le r\le 2\), and \(0\le \theta\le2\pi\) to describe the base of the cylinder. So, the integral bounds are: - \(0\le\theta\le2\pi\) - \(0\le r\le 2\) - \(0\le z\le 2\)

Write triple integral

Write the triple integral for the function \(f(x, y, z) = x^2 + y^2\) using the volume constraints given by \(D\). Convert to cylindrical coordinates for easier calculation. Cylindrical coordinates are given by: - \(x = r\cos\theta\) - \(y = r\sin\theta\) - \(z = z\) So, the triple integral becomes: $$ \int_{0}^{2\pi}\int_{0}^{2}\int_{0}^{2}(r^2\cos^2\theta + r^2\sin^2\theta)r \,dz\,dr\,d\theta $$

Calculate the triple integral

Calculate the triple integral. First, integrate with respect to \(z\): $$ \int_{0}^{2\pi}\int_{0}^{2}(r^2\cos^2\theta + r^2\sin^2\theta)r(2-0) \,dr\,d\theta \\=4\int_{0}^{2\pi}\int_{0}^{2}(r^2\cos^2\theta + r^2\sin^2\theta)r\,dr\,d\theta $$ Next, integrate with respect to \(r\): $$ 4\int_{0}^{2\pi}\left[\frac{1}{4}(r^4\cos^2\theta + r^4\sin^2\theta)\Big|_0^2\right] \,d\theta \\=4\int_{0}^{2\pi}(4\cos^2\theta + 4\sin^2\theta)\,d\theta $$ Finally, integrate with respect to \(\theta\): $$ 4\left[\int_{0}^{2\pi}(4\cos^2\theta)\,d\theta + \int_{0}^{2\pi}(4\sin^2\theta)\,d\theta\right] \\=16\pi+16\pi $$ The value of the triple integral is \(32\pi\).

Calculate average squared distance

Calculate the average squared distance by dividing the triple integral value by the total volume of the cylinder. The total volume of the cylinder is given by \(V = \pi r^2 h = \pi(2^2)(2) = 8\pi\). So, the average squared distance between the origin and points in the solid cylinder is: $$ \frac{32\pi}{8\pi} = 4 $$ Hence, the average squared distance between the origin and points in the solid cylinder is 4.

Key concepts

Cylindrical Coordinates

Cylindrical coordinates provide a useful way to describe points in three-dimensional space, especially when dealing with symmetrical shapes such as cylinders. Instead of using the Cartesian coordinates \(x, y, z\), we use the variables \(r, \ heta, zz\):

• \(x = r\cos\theta\) - here \(r\) is the radial distance from the origin in the xy-plane, and \(\theta\) is the angle between the positive x-axis and the projection of the point onto the xy-plane.
• \(y = r\sin\theta\)
• \(z = z\) - this component remains the same as in Cartesian, adding the height aspect.

Cylindrical coordinates make computations easier in problems that involve rotational symmetry about the z-axis. When integrating in cylindrical coordinates, we include an additional factor of \(r\) due to the Jacobian of the transformation, accounting for the change from Cartesian to polar coordinates in the plane.

Average Value of a Function

The average value of a function in a particular region gives a measure of its typical size over that region. For a function \(f(x, y, z)\) within a domain \(D\), the average value is calculated using the formula:

• \(\frac{1}{V} \int \int \int_D f(x, y, z) \, dV\)

where \(V\) represents the volume of the region \(D\). This approach provides a single number summarizing the behavior of \(f\) over the entire region. In our case, the function describes the squared distance from the origin over the solid cylinder. We first compute the total weighted measure from the function using a triple integral, and then divide by the volume of the cylinder to find the average.

Volume of Solids

The volume of a solid figure is crucial in various applications, particularly in calculating the average of a function over that solid. For cylinders, the volume \(V\) is computed as the product of the base area and the height:

• Base area: \(\pi r^2\), where \(r\) is the radius of the circular base.
• Height: \(h\), which is the extent of the cylinder along the z-axis.

Thus, the volume of a cylinder is given by \(V = \pi r^2 h\). Knowing the volume is critical when determining the average value of a function across the solid. It represents the total space within the solid where the calculations of averages or integrals are performed.