Final Answer:
The volume of the solid obtained by rotating the region bounded by
y=x⁴ , y=1 about the line y=9 is 171.5 cubic units.
Explanation:
To find the volume of the solid generated by rotating the region between y=x⁴ and y=1 about the line y=9, we'll use the method of cylindrical shells. The region is bounded by x=0 and x=1 (since y=x⁴ intersects y=1 at x=1). The shell method formula for volume V is V = ∫[a,b] 2πx*f(x)dx, where f(x) represents the distance between the curves being rotated. In this case, f(x) = 9 - y=x⁴ (distance from the line of rotation to the curve y=x⁴). The integral will be from 0 to 1. Thus, V = ∫[0,1] 2πx(9 - y=x⁴) dx.
Performing the integration, we get V = 2π * (∫[0,1] 9x -
dx) = 2π * [(9/2)x² - (1/6)
] evaluated from 0 to 1. Substituting the limits, V = 2π * [(9/2) - (1/6)] = 2π * (27/6) = 9π * 9/2 = 81π/2 ≈ 127.23 cubic units.
However, this volume is not the final answer because it represents the space outside the solid. To find the volume inside the solid, we subtract the volume of the cavity inside the solid, which is a cylinder with radius 1 and height 9. So, the final volume = 81π/2 - π(1)² (9) = 81π/2 - 9π = 171.5 cubic units.