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Evaluating the volume swept out by a three-dimensional (3D) object as it moves along an arbitrary path is of interest to many areas of CAD and CAM, such as mechanism design and robot path planning. This paper shows how envelope theory from differential geometry can be used to find the volumes swept out by the individual surfaces of a solid body, and how computer algebra methods may be of use to perform the computations involved. Finally, a new algorithm is presented which shows how the results of sweeping the individual surfaces of a solid body can be combined to form a new 3D model of the swept volume. This algorithm has strong resemblance to hidden line algorithms, but works in one dimension higher.