Final answer:
To find the volume of the solid generated by revolving the region bounded by the graphs of the equations about different lines, we can use the method of cylindrical shells. For each line, we set up an appropriate integral and evaluate it to find the volume.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines, we can use the method of cylindrical shells. Let's solve for each given line.
(a) y-axis: The region is bounded by the x-axis and the curve y = 2x^2. To find the volume, integrate 2πxy over the range of x from 0 to 2. The volume is ∫(0 to 2) 2πx(2x^2) dx.
(b) x-axis: The region is bounded by the curves y = 0 and y = 2x^2. To find the volume, integrate πy^2 dx over the range of y from 0 to 2. The volume is ∫(0 to 2) π(2x^2)^2 dy.
(c) y = 8: The region is bounded by the x-axis, the curve y = 2x^2, and the line y = 8. To find the volume, integrate 2πxy over the range of x from 0 to 2. The volume is ∫(0 to 2) 2πx(8-2x^2) dx.
(d) x = 2: The region is bounded by the curves y = 0, y = 2x^2, and the line x = 2. To find the volume, integrate πy^2 dx over the range of y from 0 to 2x^2. The volume is ∫(0 to 2) πy^2 dx.