Problem 41 - 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. $$x^{2}+z^{2}=4$$

Step-by-Step Solution

Answer

Question: Analyze the given equation of the cylinder, identify the coordinate axis to which it is parallel, and sketch the cylinder: x² + z² = 4 Answer: The cylinder is parallel to the y-axis and has a radius of 2.

Step 1: Identify the coordinate axis

The given equation is \(x^{2}+z^{2}=4\). Notice that the equation only involves \(x\) and \(z\) coordinates and there is no term involving the \(y\) coordinate. This means that the value of \(y\) can be anything, and thus, the cylinder is parallel to the \(y\)-axis.

Step 2: Rewrite the equation in standard form

The given equation can be written in the standard form of the equation of a cylinder as follows: $$\frac{x^{2}}{4}+\frac{z^{2}}{4}=1$$ From this, we can see that the cylinder has a radius of 2 (from \(\sqrt{4}\)).

Step 3: Sketch the cylinder

To draw the cylinder, follow these steps: 1. Draw the \(x\), \(y\), and \(z\) coordinate axes. 2. Since the cylinder is parallel to the \(y\)-axis, draw an ellipse at the base (on the \(xz\)-plane) using the radius of 2 found in step 2. 3. Draw a straight line at the edges of the ellipse pointing in the \(y\)-direction to represent the extent of the cylinder. 4. As the cylinder extends infinitely along the \(y\)-axis, you can use dotted lines to indicate that it continues in both positive and negative directions. After sketching, you should see that the cylinder has a radius of 2 and is parallel to the \(y\)-axis, as we identified earlier.