Problem 46 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The approach to finding the average value of the squared distance between the origin and points in the solid paraboloid involves converting to cylindrical coordinates, setting up triple integrals for the total volume under the paraboloid and the total squared distance, calculating both integrals, and then computing the ratio of the total squared distance to the total volume.

Step 1: Convert to cylindrical coordinates

Converting rectangular coordinates \((x, y, z)\) to cylindrical coordinates \((r, \theta, z)\), we have \(x = r\cos{\theta}\), \(y = r\sin{\theta}\), and \(z = z\). The volume element \(dV\) in cylindrical coordinates is \(r\ dr\ d\theta\ dz\).

Step 2: Set up the triple integral

To calculate the total volume under the paraboloid, we will integrate the volume element \(dV\). To calculate the total squared distance, we will integrate the squared distance function given by \(r^2 \cdot dV\). The lower and upper bounds of integration for the cylindrical coordinate variables are: \(0 \leq r \leq \sqrt{4-z}\), \(0 \leq \theta \leq 2\pi\), and \(0 \leq z \leq 4-x^2-y^2\).

Step 3: Calculate the total volume under the paraboloid

The total volume \(V\) under the paraboloid can be calculated by the triple integral: $$ V = \int_{0}^{4} \int_{0}^{2\pi} \int_{0}^{\sqrt{4-z}} r\ dr\ d\theta\ dz $$

Step 4: Calculate the total squared distance

The total squared distance \(D_{\text{total}}\) can be calculated by the triple integral: $$ D_{\text{total}} = \int_{0}^{4} \int_{0}^{2\pi} \int_{0}^{\sqrt{4-z}} r^3\ dr\ d\theta\ dz $$

Step 5: Compute the average value

Finally, we can compute the average value of the squared distance by dividing the total squared distance by the total volume. To find the actual average squared distance between the origin and points in the solid paraboloid, we must evaluate both triple integrals and compute the ratio: $$ \text{Average squared distance} = \frac{D_{\text{total}} }{V} $$