5053 -Volume by Shells AP Calculus.

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Presentation transcript:

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.”

r = the distance from the axis of rotation. Formula: Then formula is built on the surface area of a cylinder …. times …. its thickness. r = the distance from the axis of rotation. h = difference between the curves ( n )

r and h Perpendicular to x - axis

r and h Perpendicular to y - axis

Volume by Shells V = l * w * h ( 2  r * x * f (x) a) ||  shell b) c) r = ( x - 0) d) h = (f (x) - 0)

Example: Revolve the region about the y – axis. 1

Example: Rotated about x = 2

Example: Rotated about x = 3

Example: Rotated about y = 8