9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

y = (1+x²)^−1,y = 0,x = 0, and x = 2; about the y-axis

Verified step by step guidance

1. Identify the region R bounded by the curves y = 1/(1+x²), y = 0, x = 0, and x = 2. The region is revolved about the y-axis, so we will use the shell method to calculate the volume.
2. The shell method formula for volume is V = ∫[a, b] 2π(radius)(height) dx. Here, the radius is the distance from the y-axis (x), and the height is the value of the function y = 1/(1+x²).
3. Set up the integral: V = ∫[0, 2] 2π(x)(1/(1+x²)) dx. The limits of integration are from x = 0 to x = 2, as given in the problem.
4. Simplify the integrand: The integrand becomes 2πx/(1+x²). To solve this integral, consider using substitution. Let u = 1+x², so du = 2x dx.
5. Rewrite the integral in terms of u: Substitute u = 1+x² and adjust the limits of integration accordingly. Solve the resulting integral, and then substitute back to find the volume.

Key Concepts

Shell Method

The shell method is a technique for finding the volume of a solid of revolution. It involves slicing the solid into 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 horizontal strips.

Volume of Revolution

The volume of revolution refers to the volume of a three-dimensional shape created by rotating a two-dimensional area around an axis. The volume can be calculated using methods such as the disk method or the shell method, depending on the axis of rotation. Understanding how to set up the integral for volume is crucial for solving problems in calculus involving solids of revolution.

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specified limits. In the context of the shell method, the definite integral represents the total volume of the solid formed by summing the volumes of all the cylindrical shells. Mastery of evaluating definite integrals is essential for accurately determining the volume of solids of revolution.