Final answer:
To sketch the enclosed region and calculate the volume of rotation about the x-axis using the Shell Method, plot the graph of the equation y = 4 - x², determine the limits of integration, and then use the formula 2πrh to calculate the volume of rotation.
Step-by-step explanation:
To sketch the enclosed region and calculate the volume of rotation about the x-axis using the Shell Method, we first need to graph the equation y = 4 - x². This is a downward-opening parabola with its vertex at (0,4) and x-intercepts at (-2,0) and (2,0). Next, we need to determine the limits of integration. In this case, x = 0 is the lower limit of integration and y = 0 is the upper limit of integration. To calculate the volume using the Shell Method, we integrate the formula 2πrh over the given limits of integration, where r is the x-coordinate and h is the height of the shell.
Using the Shell Method, the volume of rotation about the x-axis is given by the integral:
V = ∫02 2πx(4-x²) dx
Solving this integral will give us the volume of rotation.