Question

Find the volume of rotation for the region enclosed by the graphs of y=3x²+14, y=16-2x and y=4 rotated about the line x=5?

Answer 1

To find the volume of rotation for the region enclosed by the graphs, use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of rotation for the region enclosed by the graphs of y = 3x² + 14, y = 16 - 2x, and y = 4 rotated about the line x = 5, we can use the method of cylindrical shells.

1. First, find the x-values where the three graphs intersect by setting each pair of equations equal to each other. The x-values of intersection will be the bounds of integration.
2. Next, represent the region of rotation as a series of thin cylindrical shells with radius (x - 5) and height (y2 - y3). The volume of each shell will be 2π(radius)(height).
3. Integrate the volume of each shell from the lower bound to the upper bound to find the total volume of rotation.

By following these steps, you can calculate the volume of rotation. Remember to always double-check and use correct mathematical notation in your calculations!