Question

Use the method of cylindrical shells to find the volume v generated by rotating the region bounded by the curves y = 32 - x² and y = x² about the axis x = 4?

Answer 1

To find the volume generated by rotating the region bounded by the curves y = 32 - x² and y = x² about the axis x = 4, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume generated by rotating the region bounded by the curves y = 32 - x² and y = x² about the axis x = 4, we can use the method of cylindrical shells.

1. First, we need to find the height of each cylinder. The height can be represented by the difference between the two curves: (32 - x²) - (x²) = 32 - 2x².
2. Next, we need to find the radius of each cylinder. The radius is the distance from the axis of rotation (x = 4) to the curve. Since the axis is 4 units away from the origin, the radius can be represented by 4 - x.
3. Finally, we can use the formula for the volume of a cylinder: V = 2πrh, where r is the radius and h is the height calculated in steps 1 and 2. Substituting the values, we get V = ∫(2π(4 - x))(32 - 2x²) dx, where the integral is taken from the x-coordinate of the point of intersection to the x-coordinate of the highest point of the curve y = 32 - x².