121616949-math.203 - Figure 9.10 Solid with a hole showing...

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9.3 Volume 189 giving us the usual formula for the volume of a cone. EXAMPLE 9.10 Find the volume of the object generated when the area between y = x 2 and y = x is rotated around the x -axis. This solid has a “hole” in the middle; we can compute the volume by subtracting the volume of the hole from the volume enclosed by the outer surface of the solid. In figure 9.10 we show the region that is rotated, the resulting solid with the front half cut away, the cone that forms the outer surface, the horn-shaped hole, and a cross-section perpendicular to the x -axis. 0 1 0 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Unformatted text preview: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 9.10 Solid with a hole, showing the outer cone and the shape to be removed to form the hole. ( JA ) We have already computed the volume of a cone; in this case it is π/ 3. At a particular value of x , say x i , the cross-section of the horn is a circle with radius x 2 i , so the volume of...
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