Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If a region is revolved about the x-axis, then in principle, it is possible to use the disk/washer method and integrate with respect to x or to use the shell method and integrate with respect to y.

검증된 단계별 안내

Recall the disk/washer method involves slicing the solid perpendicular to the axis of rotation, resulting in circular cross-sections (disks or washers). When revolving around the x-axis, these slices are typically horizontal, so integrating with respect to x is straightforward.

The shell method involves slicing the solid parallel to the axis of rotation, creating cylindrical shells. For revolution about the x-axis, these shells are vertical, so integrating with respect to y is natural.

To determine if both methods can be used, consider the shape of the region and the axis of rotation. If the region is described as functions of x (i.e., y = f(x)), then the disk/washer method integrating with respect to x is usually direct.

Similarly, if the region can be described as functions of y (i.e., x = g(y)), then the shell method integrating with respect to y can be applied. However, this depends on whether the region's boundaries are easily expressed in terms of y.

Therefore, in principle, when revolving around the x-axis, it is possible to use the disk/washer method integrating with respect to x or the shell method integrating with respect to y, provided the region's boundaries allow expressing the functions accordingly.

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주요 개념

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Disk/Washer Method

The disk/washer method calculates the volume of a solid of revolution by slicing perpendicular to the axis of rotation. When revolving around the x-axis, slices are vertical, and integration is typically with respect to x. Washers are used when there is a hole in the solid, subtracting the inner radius area from the outer.

Shell Method

The shell method finds volume by slicing parallel to the axis of rotation, creating cylindrical shells. For rotation about the x-axis, shells are horizontal, so integration is usually with respect to y. This method is useful when the disk/washer method is complicated or when the function is easier to express in terms of y.

Choice of Variable of Integration

The variable of integration depends on the method and axis of rotation. For rotation about the x-axis, the disk/washer method integrates with respect to x, while the shell method integrates with respect to y. Using the opposite variable often complicates the integral or is not feasible, so the statement about always being able to integrate with either variable is not always true.