Volumes_graphs - Volumes of solids of revolution Volume of a solid Let S be a solid that lies between the lines x = a and x = b If we want to find

Volumes_graphs - Volumes of solids of revolution Volume...

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Volumes of solids of revolution Volume of a solid : Let S be a solid that lies between the lines x = a and x = b . If we want to find the volume of a solid, we use the method of slicing, that is, slicing the solid with planes perpendicular to the x -axis. The strategy consist of dividing the solid into many thin “slabs” and approximating the volume of each slab by the volume of a right cylinder. If the cross-sectional area of S in the plane P x , through x and perpendicular to the x -axis, is A ( x ), where A is a continuous function, then the volume of S is V = lim n →∞ n X i =1 A ( x * i x = Z b a A ( x ) dx A solid of revolution is a solid obtained by rotating (revolving) a region in the pane about an axis. How to find the volume of a solid of revolution Case 1. If we rotate the region about the x -axis or a horizontal line, then we use V = Z b a A ( x ) dx, where A ( x ) is the cross-sectional area. Case 2. If we rotate the region about the y -axis or a vertical line, then we use V = Z d c A ( y ) dy, where A ( y ) is the cross-sectional area.

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