Volumes of Revolution The Shell Method

Volumes of Revolution The Shell Method

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

Lesson 7.3 Page 277 Exercises 1 21 odd 13

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