Section 7.3 Volume: The Shell Method. When finding volumes of solids by the disk (or washer) method we were routinely imagining our ‘slices’ under the.

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Presentation transcript:

Section 7.3 Volume: The Shell Method

When finding volumes of solids by the disk (or washer) method we were routinely imagining our ‘slices’ under the curve to be perpendicular to the axis of revolution. If we revolved around a horizontal line our integral took this form:

Section 7.3 Volume: The Shell Method When our axis of revolution was a vertical line, our ‘slices’ were horizontal disks that stacked up and down creating the following model for our volume:

Section 7.3 Volume: The Shell Method There are cases where the algebra involved in restating these functions gets more difficult than we want it to be. In those cases we have another model for what the volume will look like. For a pretty good visual, follow the link below and scroll down that page: opic/Volumes-of-Solids-of-Revolution.topicArticleId ,articleId html opic/Volumes-of-Solids-of-Revolution.topicArticleId ,articleId html

Section 7.3 Volume: The Shell Method The visual here that is helpful, I hope, is that by taking a ‘slice’ parallel to the axis of revolution you are creating concentric shells that look like rings of an onion or maybe by visualizing soup can labels. In each case we are adding areas of rectangles whose base is the perimeter of a circle and whose height is either x or y

Section 7.3 Volume: The Shell Method Consider the region formed by the curve x + y 2 = 16 and the y -axis. Revolve this region about the x-axis. We need to answer two questions to use the shell method. What is the radius? What is the height?

Section 7.3 Volume: The Shell Method Since the region is revolved around the x - axis, the radius is a y -value. The height is the difference between the curve and the y- axis (an x value). So the integral is

Section 7.3 Volume: The Shell Method One more example here: Let S be the solid formed when the region bounded by the curve y = x(x-1) 2 and the x -axis is revolved around the y -axis. Remember, ask two questions:  What is the radius?  What is the height?

Section 7.3 Volume: The Shell Method Here, the radius is an x value and the height is a y value, so the integral is