Final answer:
The volume of the solid generated by revolving the specified region about the y-axis can be found using the shell method by setting up and evaluating an integral of the form ∫(0 to 3) 2πx((7/2)x)dx.
Step-by-step explanation:
Finding the Volume Using the Shell Method
The shell method is a technique in calculus, particularly used in finding volumes of solids of revolution. The problem involves revolving the region bounded by the curves y=3x, y=-x/2, and x=3 about the y-axis. To apply the shell method, we consider a typical element (a 'shell') at a distance x from the y-axis with thickness dx. The height of this shell is the difference between the two functions, 3x and -x/2. When this shell is revolved around the y-axis, it forms a cylindrical shell.
The volume of the cylindrical shell is given by the formula V = 2π(x)(height)(thickness), where the height is y=3x - (-x/2) = (7/2)x, and the thickness is dx. We integrate this expression with respect to x from 0 to 3, the bounds given by the intersection of the linear functions and the vertical line x=3.
The integral for the volume becomes:
V = ∫(0 to 3) 2πx((7/2)x)dx
To find the volume, you'd evaluate this definite integral. Remember to simplify the integrand before integrating to make the calculation easier.