Volume Disc Washer Methods Cross Section 6 2

Volume – Disc & Washer Methods & Cross Section 6. 2

Volume – Disc Method § Solids of Revolution – if a region in the plane is revolved about a line “line-axis of revolution” § Simplest Solid – right circular cylinder or “Disc” § Volume: circular cylinder = πr 2 h

§ Disc Method – representative rectangle is perpendicular to axis (and touches axis of rotation) i) Horizontal Axis of Revolution i) Vertical Axis of Revolution

Homework § P. 430 # 1 -5, 15

Washer Method § Representative rectangle is perpendicular to the axis of revolution (does NOT touch the axis) § Solid of Revolution with a hole

Washer Method § Outer radius – inner radius

Practice Problem 1 § Find the volume of the solid generated by revolving the region bounded by the graph of y=x 3, y=1, and x=2 about the x-axis.

Practice Problem 2 Find the volume of the solid generated by revolving the region bounded by the graph of y=x 3, y=x, and between x=0 and x=1, about the y-axis.

Practice Problem 3 § Find the volume of the solid formed by revolving the region bounded by the graphs y=4 x 2 and y=16 about the line y=16.

Practice Problem 4 § Find the volume of the solid formed by revolving the region bounded by the graphs y=2 and about the line y=1.

Practice Problem 5 § Find the volume of the solid formed by revolving the region bounded by the graphs y=0, x=1 and x=4 about the line y=4

Homework § P. 430 # 11, 16, 17, 19, 23, 27, 32, 34

Cross Sections

Area of an Equilateral Triangle

Examples