Final answer:
To find the volume generated by rotating the region bounded by the given curves about the y-axis, we can use the method of cylindrical shells. The volume can be determined by integrating the product of the circumference and height of each cylindrical shell.
Step-by-step explanation:
To find the volume generated by rotating the region bounded by the curves y=3x, y=0, x=2, and x=4 about the y-axis, we can use the method of cylindrical shells.
Step 1: Determine the height of each cylindrical shell. The height will be the difference between the y-values of the curves y=3x and y=0, which is 3x - 0 = 3x.
Step 2: Determine the radius of each cylindrical shell. Since we are rotating about the y-axis, the radius will be the x-value of the curves. In this case, the radius will be x.
Step 3: Determine the differential volume of each cylindrical shell. The differential volume is given by dV = 2πrh dx, where r is the radius and h is the height.
Step 4: Integrate the differential volume from x=2 to x=4 to find the volume V. The integral expression is V = ∫2^4 2πx(3x) dx.
Step 5: Evaluate the integral to find the volume V.
Therefore, the volume V generated by rotating the region about the y-axis is given by V = ∫2^4 2πx(3x) dx.