Final answer:
To find the volume of the solid generated by revolving the plane region about the y-axis, we can use the shell method. The volume can be calculated using the formula V = 2pi∫[a,b] x*f(x)dx, where f(x) is the height of each shell. In this case, f(x) = 32 - x^(5/2).
Step-by-step explanation:
To find the volume of the solid generated by revolving the plane region about the y-axis, we can use the shell method. The shell method involves integrating the circumference of thin cylindrical shells along the y-axis. In this case, the region is bounded by the curves y = x^(5/2), y = 32, and x = 0.
First, we need to find the limits of integration. Since y = 32 is the upper boundary, the lower boundary is given by y = x^(5/2). Setting these two equations equal to each other, we get x^(5/2) = 32. Taking the fifth root of both sides, we find x = 2.
The volume can be calculated using the formula V = 2π∫[a,b] x*f(x)dx, where f(x) is the height of each shell. In this case, f(x) = 32 - x^(5/2). Substituting the limits of integration, we have V = 2π∫[0,2] x*(32 - x^(5/2))dx. Evaluating this definite integral will give us the volume of the solid.