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The Disk Method (7.2) April 17th, 2012

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I. The Disk Method Def. If a region in the coordinate plane is revolved about a line, called the axis of revolution, the resulting solid is a solid of revolution. A disk (or right circular cylinder) is the result of revolving a rectangle about an axis adjacent to one of the rectangle’s sides. axis of rotation Rectangle

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*We can apply this concept to finding the volume of a solid of revolution formed by rotating any given region in a plane by using an integral (since an integral represents the limit of an infinite number of rectangles that is used to calculate the area of the region). representative disk approximation by n disks Solid of Revolution axis of revolution representativerectangle planeregion

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The Disk Method: To find the volume of a solid of revolution with the disk method, use one of the following: Horizontal axis of revolution: Vertical axis of revolution:

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Ex. 1: Find the volume of the solid generated by revolving the region bounded by the graphs of, y=0, and x=2 about the x-axis.

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Ex. 2: Find the volume of the solid generated by revolving the region bounded by the graphs of, x=0, and y=0 about the y-axis.

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You Try: Find the volume of the solid generated by revolving the region bounded by the graphs of, x=0, and y=0 about (a) the x-axis, and (b) the y-axis.

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II. The Washer Method *If revolving a region results in a solid of revolution with a hole, we must extend the disk method with the washer method. Washer Solid of Revolution with hole axis of revolution

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The Washer Method: To find the volume of a solid of revolution generated by the revolution of a region bounded by an outer radius R(x) and an inner radius r(x), use.

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Ex. 3: Find the volume of the solid generated by revolving the region bounded by the graphs of and y=x+6 about (a) the x-axis, and (b) the line y=3.

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You Try: Find the volume of the solid generated by revolving the region bounded by the graphs of y=6-x, y=0, y=4, and x=0 about the line x=6.

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7 Applications of Integration

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