For some regions, both the washer and shell methods work well for the solid generated by revolving the region about the coordinate axes, but this is not always the case. When a region is revolved about the y-axis, for example, and washers are used, we must integrate with respect to y. It may not be possible, however, to express the integrand in terms of y. In such a case, the shell method allows us to integrate with respect to x instead. provide some insight.

Compute the volume of the solid generated by revolving the triangular region bounded by the lines 2y = x + 4, y = x, and x = 0 about

a. The x-axis using the washer method.

b. The y-axis using the shell method.

c. The line x = 4 using the shell method.

d. The line y = 8 using the washer method.