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Calculus II Chapter 6 Section 4 Volume using Shells

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David Dippel

LoneStar College - Montgomery

Revolve the region bounded by y = x2+1 y = 0 and x = 0 around the y-axis.

cross section

If we take a vertical slice

and revolve it about the y-axis

we get a cylinder.

If we add all of the cylinders together, we can reconstruct the original object.

cross section

The volume of a thin, hollow cylinder is given by:

r is the x value of the function.

h is the y value of the function.

thickness is dx.

This is called the shell method because we use cylindrical shells.

cross section

If we add all the cylinders from the smallest to the largest:

Find the volume generated when this shape is revolved about the y axis.

We can’t solve for x, so we can’t use a horizontal slice directly.

If we take a vertical slice

and revolve it about the y-axis

we get a cylinder.

Shell method:

Note:

When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

When the strip is parallel to the axis of rotation, use the shells.

When the strip is perpendicular to the axis of rotation, use disks.

Long side of Rectangle

Short Side of Rectangle

Distance to Rectangle

Long side of Rectangle

Short Side of Rectangle

p