MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
The region S is bounded by the curves x=y2 and x=2y+3. A student wants to evaluate its area using a single definite integral. Which of the following statements best explains why integrating with respect to y generally requires fewer separate integrals than integrating with respect to x?
(a) Because expressing x as a function of y produces simple polynomial integrands, whereas solving for y in terms of x introduces square roots that would force several separate dx integrals.
(b) Integrating with respect to y avoids the need to determine where the two curves intersect, whereas a dx approach requires solving for those points first.
(c) For each fixed y, the horizontal line segment inside S is bounded by exactly two curves, whereas for some x-values a vertical line intersects more than two boundary pieces.
(d) Because the line x=2y+3 is already solved for x, setting up a dy integral needs no extra algebra, but converting the parabola to y=f(x) would add extra steps and thus extra integrals.