Problem 40 - Calculus Early Transcendentals

Question

Consider the following cylinders in \(\mathbb{R}^{3}\). a. Identify the coordinate axis to which the cylinder is parallel. b. Sketch the cylinder. $$x^{2}+4 y^{2}=4$$

Step-by-Step Solution

Answer

Answer: The cylinder is parallel to the z-axis. The radii of its elliptical cross-section in the xy-plane are \(\sqrt{4}\) along the x-axis and \(\sqrt{1}\) along the y-axis.

Step 1: Identify the coordinate axis

To identify the axis that the cylinder is parallel to, we need to look at the equation $$x^2+4y^2=4$$ and determine which variable is not included in the equation, which in this case is the z-coordinate. This indicates that the cylinder is parallel to the z-axis.

Step 2: Rewrite the equation of the cylinder

To sketch the cylinder, it can be helpful to rewrite the equation in the form $$\frac{x^2}{4}+\frac{y^2}{1}=1$$ which helps us visualize the shape and size of the cylinder more easily.

Step 3: Sketch the cylinder in 3D space

Since the cylinder is parallel to the z-axis, we can imagine it as a series of circles in the xy-plane that extend along the z-axis. The rewritten equation helps us determine the size of these circles: the radius of each circle is \(\sqrt{4}\) along the x-axis and \(\sqrt{1}\) along the y-axis. Therefore, the cylinder has an elliptical cross-section in the xy-plane. Sketch the cylinder by drawing the elliptical cross-sections in the xy-plane and extending them along the z-axis. Make sure that the long radii are aligned with the x-axis and the short radii with the y-axis.