![]() Note again that f(x) and g(x) and f(y) and g(y) represent the outer and inner radii of the washers or the distance between a point on each curve to the axis of revolution.Įxample 2: Find the volume of the solid generated by revolving the region bounded by y = x 2 + 2 and y = x + 4 about the x‐axis.īecause y = x 2 + 2 and y = x + 4, you find that If the region bounded by x = f(y) and x = g(y) on, where f(y) ≥ g(y) is revolved about the y‐axis, then its volume ( V) is The volume ( V) of a solid generated by revolving the region bounded by y = f(x) and y = g(x) on the interval where f(x) ≥ g(x), about the x‐axis is If a washer is perpendicular to the y‐axis, then the radii should be expressed as functions of y. As noted in the discussion of the disk method, if a washer is perpendicular to the x‐axis, then the inner and outer radii should be expressed as functions of x. Think of the washer as a “disk with a hole in it” or as a “disk with a disk removed from its center.” If R is the radius of the outer disk and r is the radius of the inner disk, then the area of the washer is π R 2 – π r 2, and its volume would be its area times its thickness. If the axis of revolution is not a boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution, you use the washer method to find the volume of the solid. Note that f(x) and f(y) represent the radii of the disks or the distance between a point on the curve to the axis of revolution.Įxample 1: Find the volume of the solid generated by revolving the region bounded by y = x 2 and the x‐axis on about the x‐axis.īecause the x‐axis is a boundary of the region, you can use the disk method (see Figure 1). If the region bounded by x = f(y) and the y‐axis on is revolved about the y‐axis, then its volume ( V) is The volume ( V) of a solid generated by revolving the region bounded by y = f(x) and the x‐axis on the interval about the x‐axis is If a disk is perpendicular to the y‐axis, then its radius should be expressed as a function of y. If a disk is perpendicular to the x‐axis, then its radius should be expressed as a function of x. Because the cross section of a disk is a circle with area π r 2, the volume of each disk is its area times its thickness. If the axis of revolution is the boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution, then you use the disk method to find the volume of the solid. This type of solid will be made up of one of three types of elements-disks, washers, or cylindrical shells-each of which requires a different approach in setting up the definite integral to determine its volume. You can also use the definite integral to find the volume of a solid that is obtained by revolving a plane region about a horizontal or vertical line that does not pass through the plane. Volumes of Solids with Known Cross Sections.Second Derivative Test for Local Extrema.First Derivative Test for Local Extrema.Differentiation of Exponential and Logarithmic Functions.Differentiation of Inverse Trigonometric Functions. ![]() Limits Involving Trigonometric Functions.
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