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Applications of the Integral

Terms

Introduction and Summary

Areas in the Plane

Cross-Sectional Area Method  -  If A(x) is the cross-sectional area of a region perpendicular to a fixed axis at position x , defined between x = a and x = b , then the total volume of the region is

Vol = A(x)dx    

Disk Method  -  The volume of the region obtained by rotating the area under the graph of a function f (x) between x = a and x = b about the x -axis is equal to

Vol = 2Π f (x)2 dx    

This is an application of the cross-sectional area method, noting that that cross section to this surface of revolution perpendicular to the x -axis is a circle of radius f (x) .
Shell Method  -  The volume of the region obtained by rotating the area under a function f (x) between x = a and x = b about the y -axis is equal to:

Vol = 2Π xf (x)dx    

Solid of Revolution  -  The solid swept out by a region in the plane when rotated about an axis. Examples include cylinders, cones, and spheres (all considered as solids with their interiors). The volume of such a region can be computed via the disk method or the shell method.
Surface of Revolution  -  The surface swept out by a curve in the plane when rotated about an axis. Examples include the surface of a cylinder, cone, or sphere, and more generally the surface of any solid of revolution.

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