Volumes of Revolution The Shell Method Lesson 7.3 Find the volume generated when this shape is revolved about the y axis. y 4 2 x 10 x 16 9
We cant solve for x, so we cant use a horizontal slice directly. 2 If we take a vertical slice and revolve it about the y-axis we get a cylinder. y 4 2 x 10 x 16 9
3 Shell Method Based on finding volume of cylindrical shells Add these volumes to get the total volume Dimensions of the shell
Radius of the shell Thickness of the shell Height 4 The Shell Consider the shell as one of many of a dx solid of revolution f(x) f(x) g(x) x g(x)
The volume of the solid made of the sum of the shells b V 2 x f ( x) g ( x) dx a 5 Try It Out! Consider the region bounded by x = 0, y = 0, and y 8 x 2 2 2 V 2 x 8 x
0 2 dx 6 Hints for Shell Method Sketch the graph over the limits of integration Draw a typical shell parallel to the axis of revolution Determine radius, height, thickness of shell Volume of typical shell 2 radius height thickness Use integration formula b
Volume 2 radius height thickness a 7 Rotation About x-Axis Rotate the region bounded by y = 4x and y = x2 about the x-axis thickness = dy radius = y What are the dimensions needed? radius
height thickness 16 height = y V 2 y y dy 4 0 y y 4
8 Rotation About Noncoordinate Axis Possible to rotate a region around any line g(x) f(x) x=a Rely on the basic concept behind the shell method Vs 2 radius height thickness 9 Rotation About Noncoordinate Axis What is the radius?
r f(x) g(x) ax What is the height? f(x) g(x) x=c x=a What are the limits? c
The integral: a V (a x) f ( x) g ( x ) dx c 10 Try It Out Rotate the region bounded by 4 x2 , x = 0 and, y = 0 about the line x = 2 rr==22--xx 44xx22 2
Determine radius, height, limits 0 11 Try It Out Integral for the volume is 2 V 2 (2 x) (4 x 2 ) dx 0 12 Assignment
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