08-VolumeCommonShape - Volumes of Solids Using Common...

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Phong Do Volumes of Solids Using Common Geometrical Cross Sections In the disk method, the fundamental element is a disk. The volume is then found by using an integral to sum up all of the individual volumes of the disks. The disk method can be extended and adapted to cover the cases where the object is not a solid of revolution, i.e., the cross section is NOT a circular disk. Your problem may require that you use some other common shapes to define the cross sectional elements such as rectangle, triangle, etc. There is no specific formula that you can use because each problem is different, depending on the geometry of the cross section. The enclosed examples will serve to illustrate the concept. Below are some representative solids that made up of non-circular cross sections. A generic guide for problems of this type is summarized below. The area ( ) ( ) A x or A y is specific to a particular cross section that you need to know. Other than that, the differential dx or dy and the limits of integration will follow the convention used in the disk/washer method.
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