6.2 Volumes If a region in the plane is revolved about a line, the resulting solid is called a solid of revolution, the line is called the axis of revolution. The Disc Method The Washer Method the strip is perpendicular to the axis of revolution

A. Volume: The Disc Method Vertical Strip Typical disc Plane Region r Axis of Revolution x=a x= b Volume of Typical Disc Solid of Revolution where r = radius Volume of Solid

B. Volume: The Washer Method Typical Washer Vertical Strip Plane Region Solid of Revolution Axis of Revolution a b Volume of Typical Washer where = outer radius = inner radius Volume of Solid

Examples Find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disc or washer. Example One: about the y-axis Example Two: about

Solutions: Example One Graph is a parabola with vertex of (1/4,1/2) y y By disc method 1 (1/4,1/2) r x x

Solutions: Example Two y y x x 1 3 By washer method