Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

b. What is the height of a cylindrical shell at a point x in [0, 2]?

Verified step by step guidance

1. Step 1: Understand the shell method. The shell method involves calculating the volume of a solid of revolution by integrating the product of the shell radius, shell height, and 2π over the interval of interest.
2. Step 2: Identify the shell height. The shell height is the vertical distance between the upper curve y=2−x² and the lower curve y=x at a given x-value. This is calculated as the difference between the two functions: height = (2−x²)−x.
3. Step 3: Simplify the expression for the shell height. Combine like terms to express the height as a single function: height = 2−x²−x.
4. Step 4: Confirm the interval of integration. The region R is bounded in the first quadrant, and the intersection points of the curves y=2−x² and y=x occur at x=0 and x=2. Thus, the interval of integration is [0, 2].
5. Step 5: Use the shell height expression in the shell method formula. The shell height at a point x in [0, 2] is given by the simplified expression height = 2−x²−x, which will be used in the integral to compute the volume.

Key Concepts

Cylindrical Shell Method

The cylindrical shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into thin cylindrical shells, where the volume of each shell is calculated and then integrated over the interval of interest. This method is particularly useful when revolving around the y-axis, as it simplifies the integration process by using the height and radius of the shells.

Height of the Shell

In the context of the shell method, the height of a cylindrical shell at a point x is determined by the difference between the upper and lower bounding functions of the region being revolved. For the given problem, the height is calculated as the vertical distance between the curve y = 2 - x² and the line y = x, which represents the top and bottom of the region R, respectively.

Volume of Revolution

The volume of revolution refers to the volume of a three-dimensional solid formed by rotating a two-dimensional shape around an axis. The volume can be calculated using various methods, including the disk, washer, and shell methods. Understanding how to set up the integral for volume is crucial, as it involves determining the appropriate bounds and the function that describes the shape being revolved.