Final answer:
The volume of the solid obtained by rotating the region bounded by the curves y = 6x², x = 2, and y = 0 about the x-axis is 48π cubic units.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = 6x², x = 2, and y = 0 about the x-axis, we can use the method of cylindrical shells.
Here are the steps to calculate the volume:
1. Determine the interval of integration:
The region is bounded by the curves y = 6x² and y = 0. To find the x-values where the curves intersect, set the equations equal to each other: 6x² = 0.
Solving for x, we get x = 0.
Therefore, the interval of integration is from x = 0 to x = 2.
2. Set up the integral: The volume can be calculated using the formula for cylindrical shells:
V = ∫(2πx)(height)(dx)
where the height is the difference between the y-values of the two curves at a given x-value.
3. Express the curves in terms of x: The curve y = 6x² can be expressed as x = √(y/6).
4. Calculate the height: The height of the cylindrical shell at a given x-value is the difference between the y-values of the two curves. In this case, it is y = 6x² - 0 = 6x².
5. Set up the integral with the appropriate limits: The integral becomes V = ∫(2πx)(6x²)(dx) from x = 0 to x = 2.
6. Integrate to find the volume:
- Evaluating the integral, we get V = ∫12πx³ dx from x = 0 to x = 2.
- Integrating, we have V = 12π * [(x⁴)/4] from 0 to 2.
- Plugging in the limits, we get V = 12π * [(2⁴)/4] - 12π * [(0⁴)/4].
- Simplifying further, we have V = 12π * (16/4) - 0 = 48π
Thus, the volume of the solid obtained by rotating the region bounded by the curves y = 6x², x = 2, and y = 0 about the x-axis is 48π cubic units.