Chapter 6 : Applications of Definite Integrals

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本文探讨了通过切片和绕轴旋转来计算几何固体体积的方法，介绍了如何利用圆柱壳法求体积，以及平面曲线长度、质心坐标、旋转曲面面积等概念。通过定义任意基底的圆柱固体体积，引出了切片法计算非圆柱形固体体积的积分公式。

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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

$\mathbf{{\color{Red} Volume=area\times height=A\cdot h}}$

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.