Question

Find the volume V of the solid obtained by rotating the region bounded by the curves y = 3x⁶, y = 3x, x ≥ 0 about the x-axis?

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves y = 3x⁶ and y = 3x about the x-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume V of the solid obtained by rotating the region bounded by the curves y = 3x⁶ and y = 3x about the x-axis, we can use the method of cylindrical shells. This method involves integrating cross-sectional areas of infinitesimally thin cylinders. Here's how:

1. Determine the limits of integration. In this case, the curves intersect at x = 0 and x = 1.
2. Formulate the cross-sectional area of a cylinder. The radius of the cylinder is given by the distance between the curves, which is y = 3x - 3x⁶. The height of the cylinder is dx. Therefore, the cross-sectional area is given by A = 2π(3x - 3x⁶)dx.
3. Integrate the cross-sectional areas. The integral becomes V = ∫[0,1] 2π(3x - 3x⁶)dx.
4. Evaluate the integral. This involves finding the antiderivative of the function and substituting the limits of integration. The final result will give you the volume of the solid.