Let R be the region in the xy-plane between the graphs of y=e^x and y=e^-x from x=0 to x=2.
a) Find the volume of the solid generated when R is revolved about the x-axis.

Question

Let R be the region in the xy-plane between the graphs of y=e^x and y=e^-x from x=0 to x=2. 
a) Find the volume of the solid generated when R is revolved about the x-axis.

anonymous · Answer

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Please excuse the crudeness of my drawings.
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The object that you get when you revolve these equations around the x-axis looks like a sombrero with a hole down the middle.  We could take "very small" slices of this sombrero and we would end up with a bunch circles with holes in the middle.  If we want to approximate the area of this sombrero, we need to add up all the areas of circles from x=0 to x=2 but we also need to subtract out the area of the circles in the middle.