5053 -Volume by Shells

AP Calculus

Volume by Shells

V = l * w * h

( 2 r * x * f (x)

Formula:

Region may be either adjacent to or separate from the axis of rotation.

Representative Rectangle is Parallel to the axis of rotation.

“Parallel therefore shell.”

0

(thickness) * *

2 ( ) * ( ) *w

V b h w

V Lim r w f w dw

2 ( ) ( )b

a

V r w f w dw

Formula:

r = the distance from the axis of rotation.

h = difference between the curves ( n )

2 ( ) ( )b

a

V r w f w dw

Then formula is built on the surface area of a cylinder ….

times …. its thickness.

2 rh x

r and h

Perpendicular to x - axis

4y x 2 24 and 16y x x y

r and h

Perpendicular to y - axis

4y x 2 24 and 16y x x y

Volume by Shells

V = l * w * h

( 2 r * x * f (x)

a) || shell

b)

c) r = ( x - 0)

d) h = (f (x) - 0)

,x a b2 ( ) ( )

b

a

V r x f x dx

Example:

2 1 0

0 1

y x y

x x

Revolve the region about the y – axis.

1

Example:3 1

0 0 1

y x x

y x x

Rotated about x = 2

Example:2

24

y x

y x x

Rotated about x = 3

Example:3 0, 2y x y x

Rotated about y = 8