Volume – Disc & Washer Methods & Cross Sections

1. Volume – Disc & Washer Methods & Cross Sections Section 6.2

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 = πr2h

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

6. Homework • P.430 # 1-5,15

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

8. Washer Method • Outer radius – inner radius

10. Practice Problem 1 • Find the volume of the solid generated by revolving the region bounded by the graph of y=x3, y=1, and x=2 about the x-axis.

11. Practice Problem 2 Find the volume of the solid generated by revolving the region bounded by the graph of y=x3, y=x, and between x=0 and x=1, about the y-axis.

12. Practice Problem 3 • Find the volume of the solid formed by revolving the region bounded by the graphs y=4x2 and y=16 about the line y=16.

13. 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.

14. 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

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

16. Cross Sections

17. Area of an Equilateral Triangle

18. Examples