Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = x³, y = 27, and x = 0 about the x-axis using the method of cylindrical shells, integrate the expression 2πx³(27 - x³)dx from x = 0 to x = 3. The volume is 1872π cubic units.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = x³, y = 27, and x = 0 about the x-axis using the method of cylindrical shells, we need to integrate the area of the cylindrical shells.
The radius of each cylindrical shell is the distance from the x-axis to the curve y = x³, which is x³. The height of each cylindrical shell is the difference between the upper bound curve y = 27 and the lower bound curve y = x³, which is 27 - x³. The thickness of each cylindrical shell is dx.
The volume of each cylindrical shell is given by V = 2πx³(27 - x³)dx. To find the total volume, integrate this expression from x = 0 to x = 3, since the curves y = x³ and y = 27 intersect at x = 3.
∫[0 to 3] 2πx³(27 - x³)dx = π∫[0 to 3] 54x³ - 2x⁶dx = π[27x⁴ - 2/7x⁷] from 0 to 3 = π[(27(3)⁴ - 2/7(3)⁷) - (27(0)⁴ - 2/7(0)⁷)] = π[(27(81) - 2/7(2187) - (0 - 0)] = π[2187 - 2187/7] = 6π(312). The volume of the solid obtained by rotating the region about the x-axis is 1872π cubic units.