MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider two solids, Solid A and Solid B, that are both constructed with square cross sections taken perpendicular to the x-axis.
• For Solid A, the base in the xy-plane is the region bounded by the graphs of y=f(x) and y=g(x) on the closed interval [a,b] with f(x)≥g(x) for all x in the interval.
• For Solid B, the base is the region bounded by y=f(x)+k and y=g(x)+k on the same interval [a,b], where k is a positive constant.
Assume f and g are continuous on [a,b] and the areas of the cross sections are measured in square units while volumes are measured in cubic units.
(a) Describe geometrically how the bases of Solid A and Solid B are related in the coordinate plane.
(b) Explain why the sidelengths of corresponding square cross sections of the two solids are identical for every x in [a,b]. Include reasoning based on the vertical separation between the bounding curves.
(c) Determine whether the volumes of Solid A and Solid B are equal or different. Provide a clear justification using definite integrals without computing their numerical values.
(d) Suppose k is replaced by a differentiable function h(x) such that h(x)>0 for all x in [a,b]. Explain whether your conclusion in part (c) still holds. Support your answer with a concise argument.