Let region in the first quadrant be bounded above by
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let region R in the first quadrant be bounded above by y=4−x2, below by y=x2, and on the left by the y–axis. Two different solids share this same base region.
• Solid A has cross sections taken perpendicular to the x–axis that are squares.
• Solid B has cross sections taken perpendicular to the x–axis that are rectangles whose height at each x is exactly twice the width of the base at that x.
Answer the following without the aid of a calculator.
(a) For 0≤x≤2, describe explicitly how the side length of the square cross section of Solid A and the base width of the rectangular cross section of Solid B are determined from the equations of the bounding curves.
(b) Without evaluating any definite integrals, determine which solid has the greater volume. Provide a clear mathematical justification based on a comparison of the two cross–sectional areas for all admissible x–values.
(c) Identify all x–values in the interval 0≤x≤2 where the cross–sectional areas of the two solids are equal. Justify your conclusion.