Problem 1 . The region bounded by \( y=x, y=3 x \), and \( x=2 \) is revolved about the line \( x=5 \). We wish to compute the volume of the resulting solid by integrating with respect to \( x \). Which method (Washer or Shell) should be used to compute the volume? II. The region bounded by \( y=\sqrt{x}, y=0, x=1 \) and \( x=4 \) is revolved about the line \( x=8 \). We wish to compute the volume of the resulting solid using the Washer Method. Which variable should we integrate with respect to in order to compute the volume? III. The region bounded by \( y=-e^{x}, y=0, x=1 \) and \( x=2 \) is rotated about either a horizontal or a vertical line. Suppose that the volume of the resulting solid can be calculated using the Shell method, integrating with respect to \( y \). Was the axis of rotation horizontal or vertical?

When deciding whether to use the Washer or Shell method, consider the orientation of the region and the axis of rotation. For the first problem, revolving around the line \( x=5 \), the Shell method is ideal since it utilizes vertical strips which align perfectly with the axis of rotation, making calculations straightforward and visualizing the solid easier. For the second problem, when revolving the region about the line \( x=8 \) using the Washer Method, integrating with respect to \( x \) is the way to go. This method efficiently captures the thickness of the washers formed, as these washers are also oriented vertically along the x-axis, facilitating the setup of the integral to find the volume. Considering the third problem, if you're integrating with respect to \( y \) using the Shell method, the axis of rotation must be vertical. This tells us that the region extends horizontally, and the resultant shells are formed by considering vertical strips of the area being revolved around a vertical line. This setup accommodates capturing the volume through the geometry of the solid effectively!