![]() ![]() īecause the cross-sectional area is not constant, we let A ( x ) A ( x ) represent the area of the cross-section at point x. For the purposes of this section, however, we use slices perpendicular to the x -axis. Later in the chapter, we examine some of these situations in detail and look at how to decide which way to slice the solid. ![]() If we make the wrong choice, the computations can get quite messy. The decision of which way to slice the solid is very important. ![]() As we see later in the chapter, there may be times when we want to slice the solid in some other direction-say, with slices perpendicular to the y-axis. We want to divide S S into slices perpendicular to the x -axis. įigure 6.12 A solid with a varying cross-section. In the case of a right circular cylinder (soup can), this becomes V = π r 2 h. To calculate the volume of a cylinder, then, we simply multiply the area of the cross-section by the height of the cylinder: V = A The solid shown in Figure 6.11 is an example of a cylinder with a noncircular base. Thus, all cross-sections perpendicular to the axis of a cylinder are identical. A cylinder is defined as any solid that can be generated by translating a plane region along a line perpendicular to the region, called the axis of the cylinder. We define the cross-section of a solid to be the intersection of a plane with the solid. To discuss cylinders in this more general context, we first need to define some vocabulary. Although most of us think of a cylinder as having a circular base, such as a soup can or a metal rod, in mathematics the word cylinder has a more general meaning. We can also calculate the volume of a cylinder. Although some of these formulas were derived using geometry alone, all these formulas can be obtained by using integration. The formulas for the volume of a sphere ( V = 4 3 π r 3 ), ( V = 4 3 π r 3 ), a cone ( V = 1 3 π r 2 h ), ( V = 1 3 π r 2 h ), and a pyramid ( V = 1 3 A h ) ( V = 1 3 A h ) have also been introduced. The volume of a rectangular solid, for example, can be computed by multiplying length, width, and height: V = l w h. Most of us have computed volumes of solids by using basic geometric formulas. Just as area is the numerical measure of a two-dimensional region, volume is the numerical measure of a three-dimensional solid. We consider three approaches-slicing, disks, and washers-for finding these volumes, depending on the characteristics of the solid. In this section, we use definite integrals to find volumes of three-dimensional solids. In the preceding section, we used definite integrals to find the area between two curves.
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