![]() A third way this can happen is when an axis of revolution other than the x -axis x -axis or y -axis y -axis is selected. In other cases, cavities arise when the region of revolution is defined as the region between the graphs of two functions. Sometimes, this is just a result of the way the region of revolution is shaped with respect to the axis of revolution. Some solids of revolution have cavities in the middle they are not solid all the way to the axis of revolution. Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of g ( y ) = y g ( y ) = y and the y -axis y -axis over the interval around the y -axis. The following figure shows the sliced solid with n = 3. Now let P = be a regular partition of, , and for i = 1, 2 ,… n, i = 1, 2 ,… n, let S i S i represent the slice of S S stretching from x i − 1 to x i. 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. 6.2.3 Find the volume of a solid of revolution with a cavity using the washer method.6.2.2 Find the volume of a solid of revolution using the disk method. ![]() 6.2.1 Determine the volume of a solid by integrating a cross-section (the slicing method).The following depicts a side view of the triangular slice. Thus, the length of the base of an arbitrary cross sectional triangular slice is: So for that arbitrary #x#-value we have the associated #y#-coordinates #y_1, y_2# as marked on the image: In order to find the volume of the solid we seek the volume of a generic cross sectional triangular "slice" and integrate over the entire base (the circle) The grey shaded area represents a top view of the right angled triangle cross section. Consider a vertical view of the base of the object.
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