27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.


Find the volume of the solid obtained by revolving region R₂ about the y-axis.

Verified step by step guidance

1. Identify the region R₂ bounded by the curves y = 2\sqrt{x}, y = 3 - x, and the vertical line x = 3. This region lies between the curves y = 2\sqrt{x} (lower curve) and y = 3 - x (upper curve) from x = 1 to x = 3, where the two curves intersect.
2. Since the solid is formed by revolving region R₂ about the y-axis, use the method of cylindrical shells. The formula for the volume using cylindrical shells when revolving around the y-axis is: \[ V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx \] where the radius is the distance from the y-axis (which is x), and the height is the vertical distance between the two curves.
3. Determine the radius and height for the shell at a typical x between 1 and 3: - Radius = x - Height = (upper curve) - (lower curve) = (3 - x) - 2\sqrt{x}
4. Set up the integral for the volume: \[ V = 2\pi \int_{1}^{3} x \left[(3 - x) - 2\sqrt{x}\right] \, dx \]
5. Evaluate the integral by expanding the integrand and integrating term-by-term with respect to x over the interval [1, 3]. This will give the volume of the solid formed by revolving region R₂ about the y-axis.

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 methods like the disk/washer or shell method, depending on the axis of rotation and the shape of the region.

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices of the region around the y-axis. Each shell's volume is 2π(radius)(height)(thickness), making it ideal when rotating around the y-axis and the region is described in terms of x.

Setting up Integration Limits and Functions

Accurately determining the bounds of integration and the expressions for radius and height is crucial. For region R₂, bounded by y=3−x and y=2√x, and x=3, the limits are based on the intersection points of these curves, ensuring the integral correctly represents the volume.