A solid is formed so that every cross section perpendicular to the -axis is a square
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A solid is formed so that every cross section perpendicular to the x-axis is a square. The base of the solid is the entire region in the xy-plane that satisfies x2+y2≤9.
(a) Express y as an explicit function of x representing the upper semicircle and state the corresponding domain for x.
(b) Using your result from part (a), write a definite integral that gives the volume of the solid. Do not evaluate yet.
(c) Evaluate the integral from part (b) to find the exact volume of the solid. Show a complete antiderivative and clearly indicate the limits of integration.
(d) Determine the average value of the square cross-sectional area over the interval of x found in part (a). Express your answer in exact form and include units.