Final answer:
To find the volume V of the solid obtained by rotating the region bounded by the curves, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume V of the solid obtained by rotating the region bounded by the curves, we can use the method of cylindrical shells. The volume of a cylindrical shell is given by V = 2πrhΔx, where r is the radius of the shell, h is the height, and Δx is the width of the shell.
In this case, the radius is given by y, the height is given by the difference between the curves y = 2 - 1/2x and y = 0, and the width is given by dx. So we have V = 2πyx dx. To find the limits of integration, we set the two curves equal to each other and solve for x: 2 - 1/2x = 0. Solving for x, we get x = 4. Therefore, we integrate from x = 0 to x = 4.
So the volume V is given by V = ∫0⁴ 2πyx dx. Plugging in the equations for y, we have V = ∫0⁴ 2π(2 - 1/2x)x dx. Simplifying the expression, we get V = ∫0⁴ 4π - πx² dx. Evaluating the integral, we get V = [4πx - (π/3)x³]⁴0.
Substituting the limits of integration, we get V = 4π(4) - (π/3)(4)³ - (4π(0) - (π/3)(0)³). Simplifying, we get V = 16π - (64π/3). The final volume of the solid is 16π - (64π/3) cubic units.