The solid shown in Figure 2.11 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. 2.2.3 Find the volume of a solid of revolution with a cavity using the washer method.2.2.2 Find the volume of a solid of revolution using the disk method.2.2.1 Determine the volume of a solid by integrating a cross-section (the slicing method).

Volume of solid calculus calculator