Consider the region in the –plane bounded on the right by the curve and on the left…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the region in the xy–plane bounded on the right by the curve x=4−y2 and on the left by the curve x=y+2. The region is revolved about the y–axis to create a three–dimensional solid.
(a) Solve for all y–values where the two bounding curves intersect.
(b) For a fixed value of y between the intersection points, explain which of the two curves is farther from the y–axis and which is closer. Provide a clear justification using algebraic or graphical reasoning.
(c) Describe how the distances identified in part (b) determine two radii associated with a representative cross section of the solid.
(d) Explain how the y–values found in part (a) serve as the limits of integration when computing the volume of the solid. Your explanation should connect the region’s geometry to the structure of the definite integral.