Question

Find the volume V of the solid obtained by rotating the region bounded by the curves y = 6 - 5x², y = 0, x = 0, x = 1 about the x-axis.

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves y = 6 - 5x², y = 0, x = 0, x = 1 about the x-axis, you can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves y = 6 - 5x², y = 0, x = 0, x = 1 about the x-axis, we can use the method of cylindrical shells.

1. Determine the limits of integration. Since the curves intersect at x = 0 and x = 1, the limits of integration are 0 and 1.
2. Find the height of each cylindrical shell. The height of each shell is given by the difference between the upper and lower curves: h = (6 - 5x²) - 0 = 6 - 5x².
3. Find the radius of each cylindrical shell. The radius is simply x.
4. Write the formula for the volume of a cylindrical shell: V = 2πrh*dx
5. Integrate the formula from step 4 with respect to x and evaluate it between the limits of integration.

Using this method, the volume of the solid obtained by rotating the region bounded by the curves y = 6 - 5x², y = 0, x = 0, x = 1 about the x-axis is the integral of 2π(6 - 5x²)x*dx evaluated from 0 to 1.