Volume Shell Method Notes, Lecture notes of Differential and Integral Calculus

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APPLICATIONS OF INTEGRATION

Volumes by

Cylindrical Shells

APPLICATIONS OF INTEGRATION

Learning Objective:

• Learn how to apply the method of cylindrical

shells to find out the volume of a solid.

If we slice perpendicular to the y - axis, we get a washer. ▪ However, to compute the inner radius and the outer radius of the washer, we would have to solve the cubic equation y = 2 x 2

• x 3 for x in terms of y.

VOLUMES BY CYLINDRICAL SHELLS

Fortunately, there is a method—the method of cylindrical shells—that is easier to use in such a case.

VOLUMES BY CYLINDRICAL SHELLS

Its volume V is calculated by subtracting the volume V 1 of the inner cylinder from the volume of the outer cylinder V 2 .

CYLINDRICAL SHELLS METHOD

Thus, we have: 2 1 2 2 2 1 2 2 2 1 2 1 2 1 2 1 2 1 ( ) ( )( ) 2 ( ) 2 V V V r h r h r r h r r r r h r r h r r

= − = − = − = + −

= −

CYLINDRICAL SHELLS METHOD

The equation can be remembered as: V = [circumference] [height] [thickness]

CYLINDRICAL SHELLS METHOD

V = 2  rh  r

Now, let S be the solid obtained by rotating about the y - axis the region bounded by y = f ( x ) [where f ( x ) ≥ 0], y = 0 , x = a and x = b , where b > a ≥ 0.

CYLINDRICAL SHELLS METHOD

The rectangle with base [ x i - 1 , x i ] and height is rotated about the y - axis. ▪ The result is a cylindrical shell with average radius , height , and thickness ∆ x. ( ) i f x ( ) i f x i

x

CYLINDRICAL SHELLS METHOD

Thus, by Formula 1, its volume is calculated as follows: (2 )[ ( )]

i i i

V =  x f x  x

CYLINDRICAL SHELLS METHOD

The approximation appears to become better as n →∞. However, from the definition of an integral, we know that: 1 lim 2 ( ) 2 ( ) n b i i n a i  x f x x  x f x dx → =  =  (^) 

CYLINDRICAL SHELLS METHOD

Thus, the following appears plausible. ▪ The volume of the solid obtained by rotating about the y - axis the region under the curve y = f ( x ) from a to b , is: where 0 ≤ a < b 2 ( ) b a V =  xf x dx  CYLINDRICAL SHELLS METHOD Formula 2

This type of reasoning will be helpful in other situations—such as when we rotate about lines other than the y - axis.

CYLINDRICAL SHELLS METHOD

Find the volume of the solid obtained by rotating about the y - axis the region bounded by y = 2 x

• x

and y = 0. 16π/ CYLINDRICAL SHELLS METHOD Example 1