Calculus 1 — 28.5: Washer Method with Non-Axis Revolution скачать в хорошем качестве

Calculus 1 — 28.5: Washer Method with Non-Axis Revolution

When you revolve a region around a line like y = 3/2 instead of the x-axis, the radii change in ways that trip up most students. This video walks through the full setup, expansion, and integration for the region between y = x and y = x³ on [0, 1] revolved around y = 3/2, arriving at a final volume of 47π/84. Key concepts covered: • Why the standard washer formula changes when the axis of revolution is not y = 0 • Identifying outer vs. inner radius by distance to the line of revolution, not position on the graph • The key misconception: the lower curve (y = x³) is farther from y = 3/2, making it the outer radius • Setting up V = π∫₀¹ [(3/2 − x³)² − (3/2 − x)²] dx • Expanding squared binomials and canceling constant terms before integrating • Term-by-term integration and combining fractions over a common denominator of 84 • A reusable four-step checklist for any washer problem with a non-axis line of revolution • Challenge problem: revolving the same region around y = −1 and verifying the principle still holds ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 1 Lecture 5.2:  Volume of Solids ...