Final Answer:
The volume of the solid formed by rotating the region enclosed by y = e⁵x⁴, y = 0, x = 0, x = 0.5 about the y-axis is approximately 0.023 cubic units.
Step-by-step explanation:
To find the volume of the solid formed by rotating the given region about the y-axis, we'll use the method of cylindrical shells. The region bounded by y = e⁵x⁴, y = 0, x = 0, and x = 0.5 will be revolved around the y-axis.
First, determine the limits of integration by setting up the integral with respect to y, which in this case are y = 0 and y = e⁵(0.5)⁴. We integrate with respect to y because the region is bounded by y-values.
The integral setup for the volume using cylindrical shells is V = 2π∫[a to b] x*f(x) dx, where f(x) represents the function that bounds the region, and x is the variable of integration.
The function that bounds the region is x = (y^(1/5))/(e^(4/5)). The integral becomes V = 2π∫[0 to e⁵(0.5)⁴] ((y^(1/5))/(e^(4/5))) * y dy.
Solving the integral gives V ≈ 0.023 cubic units after the calculations. This represents the volume of the solid generated by rotating the region about the y-axis.
In summary, utilizing the method of cylindrical shells and the integral setup with appropriate bounds, the volume of the solid formed by rotating the specified region around the y-axis is approximately 0.023 cubic units.