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  • Chapter 6 : Applications of Definite Integrals

     

    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

    mathbf{{color{Red} Volume=area	imes height=Acdot h}}

    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. 


    6.2 Volumes by Cylindrical Shells



    6.3 Lengths of Plane Curves


    6.4 Moments and Centers of Mass

    The coordinates of the centroid of a differentiable plane curve are 

    ar x=frac{int yds}{length}             ar y=frac{int xds}{length}


    6.5 Areas of Surfaces of Revolution and the Theorems of Pappus

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  • 原文地址:https://www.cnblogs.com/chintsai/p/11829253.html
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