Final answer:
To calculate the volume V generated by rotating the region bounded by y = x⁵, y = 0, and x = 2 about the y-axis, we solve y = x⁵ for x in terms of y, and then set up the integral using the disk method from the lower bound 0 to the upper bound 32 on y, resulting in π∫_0^32 y⁴⁰ dy.
Step-by-step explanation:
To set up the integral for the volume V generated by rotating the region bounded by the curves y = x⁵, y = 0, and x = 2 about the y-axis, we use the disk method as the region is rotated around the y-axis, which is perpendicular to the x-axis. The integral is a function of y since we are rotating around the y-axis. Therefore, we need to express x in terms of y.
First, solve y = x⁵ for x in terms of y by taking the fifth root: x = y¹⁵. The limits of integration are from y = 0 (the lower bound) to y = 2⁵ (the upper bound, given by substituting x = 2 into the equation y = x⁵). The volume integral is:
V = ∫ (πx²) dy = ∫_0^32 π(y¹⁵) ² dy = π ∫_0^32 y²⁵ dy
This results in:
V = π ∫_0^32 y⁴⁰ dy
Here, we have the radius of the disk as r(y) = y¹⁵ and thus the area of the disk is πr(y)² = π(y¹⁵)². Proceed by evaluating this definite integral to get the volume.