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Example 1. Set up an integral that gives the surface area of revolution about the x axis of the curve. y = x2 (1.5.3) (1.5.3) y = x 2. from 2 to 3. Solution. We find. (dy dx)2 = (2x)2 = 4x2. (1.5.4) (1.5.4) ( d y d x) 2 = ( 2 x) 2 = 4 x 2. Now use the area formula: A = 2π∫3 2 x2 1 + 4x2− −−−−−√ dx. (1.5.5) (1.5.5) A = 2 π ∫ ...
Volumes of Revolution. Rotation About the x-axis. Integration can be used to find the area of a region bounded by a curve whose equation you know. If we want to find the area under the curve y = x 2 between x = 0 and x = 5, for example, we simply integrate x 2 with limits 0 and 5. Now imagine that a curve, for example y = x 2, is rotated around ...
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In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can use this method on the same kinds of solids as the disk method or the washer method; however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution.
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Surfaces of revolution and solids of revolution are some of the primary applications of integration. A two-dimensional curve can be rotated about an axis to form a solid, surface or shell. Use Wolfram|Alpha to accurately compute the volume or area of these solids. Examples of the methods used are the disk, washer and cylinder method.
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