Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis. y=3+2x−x2,x+y=3

Answer 1

To find the volume generated by rotating the region bounded by the curves y=3+2x-x^2 and x+y=3 about the y-axis using the method of cylindrical shells, follow these steps.

Step-by-step explanation:

To find the volume generated by rotating the region bounded by the curves y=3+2x-x^2 and x+y=3 about the y-axis using the method of cylindrical shells, we need to set up the integral expression.

Step 1: Find the intersection points of the curves by solving the system of equations:

y=3+2x-x^2 and x+y=3

Step 2: Determine the limits of integration by finding the y-values of the intersection points.

Step 3: Set up the integral expression using the formula for the volume of a cylindrical shell.

Step 4: Evaluate the integral to find the volume.