Consider the region S in the xy-plane bounded on the left by the curve x=y2, on the right by the curve x=4−y2, and above and below by their points of intersection.
The curves intersect at y=−2 and y=2.
(a) Write a complete set of definite integral(s) that give the area of S when vertical slices are used.
(b) Write a single definite integral that gives the area of S when horizontal slices are used.
(c) Evaluate the definite integral(s) from part (a) and from part (b), showing all steps. Confirm that both methods produce the same numerical area.
(d) Compare the algebraic complexity of the two approaches. Cite at least two specific procedural steps that differ in difficulty between the two methods.