Problem 45 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: Set up the squared distance function

For any point (x,y,z) in 3-dimensional space, the squared distance from the origin (0,0,0) can be computed as \(x^2+y^2+z^2\). Since we are concerned with points within the cylinder, we will use this squared distance function.

Step 2: Set up the integral for average value

We want to calculate the integral of our distance function inside the cylinder and then divide by the volume of the cylinder. First, let's use cylindrical coordinates (r,θ,z) as they are more suitable for this problem. The transformation from Cartesian coordinates is given by: x = r cos θ, y = r sin θ, z = z Now, we can rewrite the squared distance function as: \(r^2\cos^2\theta + r^2\sin^2\theta + z^2 = r^2 + z^2\) (since \(\cos^2\theta + \sin^2\theta = 1\)). To calculate the integral, we need to transform the volume element in Cartesian coordinates, dV = dx dy dz, to cylindrical coordinates. In cylindrical coordinates, the volume element is given by dV = rdr dθ dz. The required integral for the squared distance function is: \(\displaystyle\int_{0}^{2\pi}\int_{0}^{2}\int_{0}^{2} (r^2 + z^2) rdr d\theta dz\)

Step 3: Calculate the integral

Let's calculate the integral step by step using the following steps: 1. Integrate with respect to r: \(\displaystyle\int_{0}^{2} r(r^2 + z^2) dr = \frac{1}{4}(r^4 + 2r^2z^2)|_{0}^{2} = 4 + 4z^2\) 2. Integrate with respect to z: \(\displaystyle\int_{0}^{2}(4 + 4z^2) dz = (4z + \frac{4}{3}z^3)|_{0}^{2} = 16\) 3. Finally, integrate with respect to θ: \(\displaystyle\int_{0}^{2\pi} 16 d\theta = 16\theta|_{0}^{2\pi} = 32\pi\) Now, let's calculate the volume of the cylinder. The volume of a cylinder is given by \(\pi r^{2}h\), where r is the radius and h is the height. In this case, r = 2 and h = 2, so the volume is \(8\pi\). The average value is obtained by dividing the integral by the volume of the cylinder: Average value \(= \frac{32\pi}{8\pi} = 4\). The average squared distance between the origin and points within the solid cylinder is 4.