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Published byJanice Gornall Modified over 3 years ago

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Volumes using washers

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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…

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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?

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The disks now have holes in them, making them more like washers. If we use vertical slices Rotate the region about the x-axis 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.

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When we subtract, we get… The volume of the disk is R 2 R r The volume of the hole is r 2 outer radius inner radius dx

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The region bounded by and is revolved about the y-axis. Find the volume. The volume of the washer is: outer radius inner radius

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If the same region is rotated about the line x = 2 : The outer radius is: R The inner radius is: r

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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. The washer method formula is: 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.

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The washer method formula is: 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? R r

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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: outer radius inner radius thickness of slice Cylinder First, the cylinder on the bottom Everything above that will require washers…

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outer radius inner radius thickness of slice Cylinder

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Volume. Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2.

Volume. Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2.

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