Question

use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. xy=5,x=0,y=5,y=7

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves xy=5, x=0, y=5, and y=7 about the x-axis using the method of cylindrical shells, follow these steps:

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves xy=5, x=0, y=5, and y=7 about the x-axis using the method of cylindrical shells, follow these steps:

1. First, determine the bounds of the region of rotation. In this case, the region is bounded by the lines x=0, y=5, and y=7.
2. Next, find the height of each cylindrical shell. The height can be calculated as the difference between the upper boundary (y=7) and the lower boundary (y=5).
3. Then, calculate the radius of each cylindrical shell. Since the region is rotated about the x-axis, the radius is equal to the x-coordinate of each point on the curve xy=5. In this case, the radius is equal to 5/y.
4. Finally, use the formula for the volume of a cylindrical shell, V = 2πrhΔx, to calculate the volume of each shell. Integrate this equation from x=0 to x=5 to obtain the total volume.

By following these steps, you can find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.