Let R be the region in the first quadrant bounded above by the line y=4−x, below by the line y=x, and below by the x-axis. The solid S is generated when R is revolved about the x-axis.
(a) Sketch R on the coordinate plane and clearly label all points where the bounding curves intersect the x-axis and each other.
(b) Determine the x-coordinate at which the two oblique lines intersect.
(c) Write, but do not yet evaluate, the definite integral or integrals that represent the volume of S.
(d) Evaluate the integral expression(s) from part (c) to obtain the exact volume of S.