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Volumes using washers

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Volumes using washers

Now that you have successfully designed a 4 by 4 meter nose cone, your boss brings to you a larger nose cone that is 16 meters long by 8 meters wide.

In order to safely ship it, you need to cover it with a layer of insulation exactly as thick as shown in the gray area below:

What you know about the design is that the curvature of the nose cone and the insulation can be given by…

Now that you have successfully designed a 4 by 4 meter nose cone, your boss brings to you a larger nose cone that is 16 meters long by 8 meters wide.

In order to safely ship it, you need to cover it with a layer of insulation exactly as thick as shown in the gray area below:

How many cubic meters of insulation will it take to make this layer?

Rotate the region about the x-axis

If we use vertical slices

The “disks” now have holes in them, making them more like “washers”.

We find the volume of a washer by considering it to be two disks; an outer and inner disk. We find the volume by subtracting the volume of the hole from the volume of the disk.

R

The volume of the disk is R2

r

The volume of the hole is r2

When we subtract, we get…

dx

outer

radius

inner

radius

The volume of the washer is:

The region bounded by

and is revolved about the y-axis.

Find the volume.

outer

radius

inner

radius

r

R

If the same region is rotated about the line x=2:

The outer radius is:

The inner radius is:

The washer method formula is:

This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.

Where R is the distance from the axis of rotation to the outer curve and r is the distance from the axis of rotation to the inner curve.

The washer method formula is:

r

R

Where R is the distance from the axis of rotation to the outer curve and r is the distance from the axis of rotation to the inner curve.

Since the region being rotated is one bound by two curves, could we also consider R to be the upper curve and r to be the lower curve?

Find the volume of the region bounded by , , and revolved about the y-axis.

We can use the washer method if we split it into two parts:

First, the cylinder on the bottom

Everything above that will require washers…

Cylinder

inner

radius

outer

radius

thickness

of slice

Cylinder

inner

radius

outer

radius

thickness

of slice

p