A region in the first quadrant is bounded by the curves and
MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A region in the first quadrant is bounded by the curves x=y2 and x=6y. The region is revolved about the y-axis to generate a solid. Which of the following statements best justifies choosing y as the variable of integration when computing the volume?
(a) Because the axis of rotation is vertical, the slices must be vertical, so integrating with respect to x is required.
(b) Since the equations of the curves are already written as x=f(y), integrating with respect to x avoids additional algebra.
(c) Horizontal slices at fixed y produce circular cross-sections whose radii are measured horizontally from the y-axis, so integrating with respect to y directly accumulates their areas.
(d) Using y as the variable would require two separate integrals because the outer radius changes, so integrating with respect to x is simpler.