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.


Use the shell method to find an integral, or sum of integrals, that equals the volume of the solid obtained by revolving region R₃ about the line x=3. Do not evaluate the integral.

Verified step by step guidance

1. Identify the region R₃ bounded by the curves y = 2\sqrt{x}, y = 3 - x, and the vertical line x = 3. From the graph, R₃ lies between the curves y = 2\sqrt{x} (upper boundary) and y = 3 - x (lower boundary) from their intersection point to x = 3.
2. Determine the axis of revolution, which is the vertical line x = 3. Since we are revolving around a vertical line, the shell method involves integrating with respect to y or x, using cylindrical shells parallel to the axis of revolution.
3. Set up the shell radius and height. The radius of a shell at a point x is the horizontal distance from x to the line x = 3, which is (3 - x). The height of the shell is the vertical distance between the two curves, which is the difference between the upper curve y = 2\sqrt{x} and the lower curve y = 3 - x.
4. Find the limits of integration by determining the x-values where the two curves intersect. Solve 2\sqrt{x} = 3 - x to find the intersection points, which will serve as the bounds for the integral.
5. Write the integral for the volume using the shell method formula: \(V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx = 2\pi \int_{a}^{b} (3 - x) \left(2\sqrt{x} - (3 - x)\right) \, dx\), where a and b are the intersection points found in step 4.

Key Concepts

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. Each shell's volume is approximated by 2π(radius)(height)(thickness), where the radius is the distance from the axis of rotation, the height is the function value, and the thickness is a small change in the variable of integration.

Setting up the Integral with Respect to y or x

Choosing the variable of integration depends on the axis of rotation and the region's boundaries. For rotation about a vertical line (x=3), integrating with respect to y or x requires expressing the radius and height in terms of that variable, ensuring the limits correspond to the region R₃.

Understanding the Region R₃ and Its Boundaries

Region R₃ is bounded by y = 2√x, y = 3 - x, and x = 3. Identifying the intersection points and the limits of integration is crucial to correctly describe the height and radius of shells. This ensures the integral accurately represents the volume of the solid formed by revolving R₃ about x=3.