Final answer:
The volume of the solid created by rotating the region bounded by y=x^5, y=1, and the y-axis around the x-axis is obtained using the disc method, resulting in π cubic units.
Step-by-step explanation:
The question asks to find the volume of the solid obtained by rotating the region in the first quadrant bounded by y = x^5, y = 1, and the y-axis around the x-axis. This is a typical calculus problem involving the use of the disc method or the washer method, depending on whether the region is solid or has a hole, respectively.
Since we need to rotate around the x-axis, we will use the disc method where the volume (V) is given by the integral V = π ∫ (outer radius)^2 - (inner radius)^2 dx, over the interval from x=0 to where y=1 intersects y=x^5. Since y=1 intersects y=x^5 at x=1, we can find this volume by evaluating the integral V = π ∫_0^1 (1)^2 dx, which simplifies to V = π [x]∫_0^1. Thus, the volume is V = π (1)^3 - (0)^3 = π cubic units.
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