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MathCalculus

Shell Method: Volume Under y = 1 − x² About the y-Axis

Use cylindrical shells to integrate the region under 1 − x² over [0,1] about the y-axis and verify its volume of π/2.

Question

Use cylindrical shells to calculate the volume obtained by rotating the region under y=1−x2y=1-x^2 and above y=0y=0, for 0≤x≤10\le x\le1, about the yy-axis.

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Evidence boundary

This solution uses OpenStax Calculus Volume 1, Section 6.3, Exercise 120, including the shared instruction specifying rotation about the y-axis. All boundaries are supplied by that exercise. The volume is a mathematical calculation in cubic units; no measurement or experimental observation is inferred.