Problem 38 - Calculus Early Transcendentals

Question

Identify and sketch the following sets in spherical coordinates. $$\\{(\rho, \varphi, \theta): \rho=2 \sec \varphi, 0 \leq \varphi<\pi / 2\\}$$

Step-by-Step Solution

Answer

Based on the given equation in spherical coordinates, ρ = 2 sec φ, determine the corresponding Cartesian equation and sketch the set in 3D space. Solution: 1. Convert the given equation to Cartesian coordinates, which is z = 2. 2. Sketch the set by drawing a horizontal plane parallel to the xy-plane at a height of 2 units above the xy-plane. Label this plane as z = 2. 3. Shade only the region of the z = 2 plane within the upper hemisphere since the bounds for φ are between 0 and π/2. Note: The solution set represents the part of a horizontal plane parallel to the xy-plane at a height of 2 units above the xy-plane and lies within the upper hemisphere.

Step 1: Convert to Cartesian Coordinates

Using the given equation in spherical coordinates and the general formulas for conversion, we can convert the set to Cartesian coordinates: $$\rho = 2 \sec \varphi$$ Using the fact that \(\sec \varphi = \frac{1}{\cos \varphi}\), we have: $$\rho = 2 \frac{1}{\cos \varphi}$$ Now, we know that \(z = \rho \cos \varphi\), so substituting the expression for \(\rho\) gives us: $$z = 2 \frac{1}{\cos \varphi} \cos \varphi$$ $$z = 2$$ Thus, the Cartesian equation for the given set is \(z = 2\).

Step 2: Sketch the Set

Now that we have converted the set to Cartesian coordinates, we can sketch it. The equation \(z = 2\) represents a horizontal plane parallel to the xy-plane at a height of 2 units above the xy-plane. To sketch this, draw the standard xyz-coordinate axes. Then, draw a horizontal plane parallel to the xy-plane at a height of 2 units above the xy-plane. Label this plane as \(z = 2\). This plane represents the given set in spherical coordinates. Since the bounds for \(\varphi\) are between \(0\) and \(\frac{\pi}{2}\), it means that only the region of the plane that lies within the upper hemisphere (i.e., above xy-plane) is relevant for this exercise. To show this, shade the region of the xy-plane below the plane \(z = 2\) and the region of the \(z = 2\) plane within the upper hemisphere to emphasize that only the latter is part of the solution set.