Solids of Revolution Shell Method

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Presentation transcript:

Solids of Revolution Shell Method Section 7-3-A Solids of Revolution Shell Method

Cylindrical Shell Method When an area between two curves is revolved about an axis a solid is created. This solid is the sum of many, many concentric cylinders. Volume = ∑circumference ∙ height ∙thickness

The Shell Method Consider a representative rectangle as shown where w is the width of the rectangle, h is the height of the rectangle and rotation is About the y-axis (vertical) http://www.mathdemos.org/mathdemos/shellmethod/

Shell Method: Horizontal Axis of Revolution Slice parallel to the revolution axis Vertical Axis of Revolution Slice parallel to the revolution axis

Shell Method: V = 2prhw Horizontal Axis of Revolution Integrate in terms y Vertical Axis of Revolution Integrate in terms x Width Radius Height

Two Functions: The volume of the solid of revolution of the region bounded by f(x) and g(x) is given by:

1) Find the volume of the region bounded by 1) Find the volume of the region bounded by the x – axis, x = 1, and x = 4 revolve about the y – axis. 1 4

2) Find the volume of the solid formed by revolving the region bounded by the graph of , the x – axis, and x = 9 about the y - axis. 9

3) Find the volume of the solid formed by 3) Find the volume of the solid formed by revolving the region bounded by the graph of x = y3 and x = y from y = 0 to y = 1about the x - axis. Need in terms of y 1

4) Find the volume of the solid formed by 4) Find the volume of the solid formed by revolving the region bounded by the graph of x = y3 and x = y from y = 0 to y = 1 about the line y = -1. 1 y = -1

Homework Page 474 # 1-9, & 15-19, set up and integrate using math 9 # 23 find the volume using the shell method