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cylinder. inner radius. outer radius. thickness of slice. Here is another way we could approach this problem:. If we take a vertical slice. and revolve it about the y-axis. we get a cylindrical shell.

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## outer radius

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**cylinder**inner radius outer radius thickness of slice**Here is another way we could approach this problem:**If we take a vertical slice and revolve it about the y-axis we get a cylindrical shell. If we add all of the cylindrical shells together, we can reconstruct the original object. Imagine cylinders within cylinders in which all you wanted was the volume only of the shell not the inside of the cylinder**Here is another way we could approach this problem:**cross section If we take a vertical slice and revolve it about the y-axis we get a cylindrical shell. If we add all of the cylindrical shells together, we can reconstruct the original object.**Picture this cylinder like you would a soup label and**imagine it being stretched out flat like this… cross section The volume of a thin, hollow cylinder is given by: C ? r is the x coordinate. h is the y coordinate. thickness is dx.**Notice that your answers were the same with both methods**Remember that you’re finding the same volume just by different methods; one by rotating horizontal rectangles and the other by rotating vertical ones Another way to look at shells would be to look at the layers of paper formed by the pages of a magazine if you were to roll the magazine up into a cylindrical shape.**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.**If we used horizontal rectangles, we’d get…**WASHERS…but wait! R and r are the same curve so this won’t work. Since we can’t solve for x, we can’t use a horizontal slice directly.**If we take vertical slices**Shell method: and revolve them about the y-axis we get cylinders. Volume of the shell = Remember: we don’t want the volume of the cylinder, just the shell of the cylinder**Find the volume when rotating this region about the y-axis**r = x h = y = x2**Find the volume when rotating this region about the x-axis**r = y h =**Find the volume when rotating this region about the line x =**2 r = 2 – x h = y = x2**When the strip is parallel to the axis of rotation, use the**shell method. When the strip is perpendicular to the axis of rotation, use the disk or washer method. y-axis The height is the distance from the upper curve to the lower curve. f (x) When rotating about the y-axis, the radius is x. h r**x = a**The height is the distance from the upper curve to the lower curve. f (x) h When rotating about the line x = a, the radius is a – x if a > x (if the region is to the left of a) r x = a f (x) When rotating about the line x = a, the radius is x – a if x > a (if the region is to the right of a) h The same applies when rotating about the y-axis except that everything needs to be in terms of y r p

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