43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.


The region bounded by the curve y = 1+√x, the curve y = 1−√x, and the line x=1 is revolved about the y-axis. Find the volume of the resulting solid by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your answers agree.

Verified step by step guidance

1. First, understand the region bounded by the curves: \(y = 1 + \sqrt{x}\), \(y = 1 - \sqrt{x}\), and the vertical line \(x = 1\). Sketching the region helps visualize the area to be revolved around the y-axis.
2. For part (a), integrating with respect to \(x\): Since the solid is formed by revolving around the y-axis, use the shell method. The shell radius is the distance from the y-axis, which is \(x\), and the shell height is the difference between the upper and lower curves, \(\left( (1 + \sqrt{x}) - (1 - \sqrt{x}) \right) = 2\sqrt{x}\). The volume integral is then \(V = \int_0^1 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\).
3. Set up the integral for part (a): \(V = \int_0^1 2\pi x \cdot 2\sqrt{x} \, dx = \int_0^1 4\pi x \sqrt{x} \, dx\). Simplify the integrand before integrating.
4. For part (b), integrating with respect to \(y\): Express \(x\) in terms of \(y\) from the given curves. From \(y = 1 + \sqrt{x}\), solve for \(x\) to get \(x = (y - 1)^2\). Similarly, from \(y = 1 - \sqrt{x}\), \(x = (1 - y)^2\). The region in \(y\) goes from \(y = 0\) to \(y = 2\) (since at \(x=1\), \(y\) ranges between \(1 - 1 = 0\) and \(1 + 1 = 2\)).
5. Use the disk/washer method for part (b): The radius of the washers is the horizontal distance from the y-axis to the outer curve minus the inner curve, which is \(R(y) = 1\) (since \(x=1\) is the boundary) and \(r(y) = (1 - y)^2\) or \((y - 1)^2\) depending on the side. Set up the volume integral as \(V = \int_0^2 \pi \left( R(y)^2 - r(y)^2 \right) dy\). Evaluate this integral to find the volume.

Key Concepts

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The volume can be computed using integral calculus by summing infinitesimal cross-sectional areas perpendicular to the axis of rotation.

Disk/Washer and Shell Methods

These are two common techniques for finding volumes of solids of revolution. The disk/washer method slices perpendicular to the axis, using circular cross-sections, while the shell method uses cylindrical shells by slicing parallel to the axis. Choosing the appropriate method simplifies integration.

Integration with Respect to Different Variables

Integrating with respect to x or y changes the approach to setting up the integral, affecting limits and expressions for radii or heights. Understanding how to express the region and boundaries in terms of each variable is essential to ensure consistent volume calculations.