how do i solve this using the washer and shell methods

Question

In Exercises 1-4 find the volume of the solid generated by revolving the region bounded by y=2-x, x=0, y=0 about the indicated axis, using the indicated element of volume. 1) x-axis (disks) 2)y-axis(disks) 3)x-axis(shells) 4)y-axis(shells)

Answer 1

Above is the graph of y=2-x or x=2-y, which creates a symmetry evident when the integrations are carried out.

Above is the integration for all parts (1) to (4) of the question.

The solid is a cone with height=base=2, volume=⅓πr²h=8π/3, confirming the integrations.

Whether dx or dy is used in the integration is usually determined by which axis is used as the “spindle” for rotation.