6.1 Volumes by Slicing and Rotation About an Axis

In this section we define volumes of solids whose cross-sections are plane regions. A cross-section of a solid S is the plane region formed by intersecting S with a plane (Figure 6.1).

Suppose we want to find the volume of a solid S like the one in Figure 6.1. We begin by extending the definition of a cylinder from classical geometry to cylindrical solids with arbitrary bases (Figure 6.2). If the cylindrical solid has a known base area A and height h, then the volume of the cylindrical solid is

This equation forms the basis for defining the volumes of many solids that are not cylindrical by the method of slicing. If the cross-section of the solid S at each point in the interval [a, b] is a region R(x) of area A(x), and A is a continuous function of x, we can define and calculate the volume of the solid S as a definite integral in the following way. 

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6.2 Volumes by Cylindrical Shells

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6.3 Lengths of Plane Curves

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6.4 Moments and Centers of Mass

The coordinates of the centroid of a differentiable plane curve are 

             

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6.5 Areas of Surfaces of Revolution and the Theorems of Pappus