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Chapter 6: Problem 5

Why is the disk method a special case of the general slicing method?

Short Answer

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In summary, the disk method can be considered a special case of the general slicing method because it calculates the volume of a solid of revolution by summing the areas of infinitesimally thin circular disks, which is a specific scenario of the general slicing method. The general slicing method calculates the volume of any solid by summing the areas of its cross-sectional slices, while the disk method does this specifically for solids generated by revolving a curve about an axis. Thus, the disk method is essentially the general slicing method with a unique circular cross-sectional area function taken into account.

Step by step solution

01

Explain the General Slicing Method

The general slicing method is used to find the volume of a solid by integrating the area of its cross-sectional slices. In this method, we imagine taking a series of infinitesimally thin planar slices perpendicular to the axis of rotation, and summing up their areas to find the volume of the solid. Mathematically, the volume can be expressed as: $$ V = \int_a^b A(x) dx $$ where V is the volume of the solid, A(x) is the area of a single slice at position x, and the integration is performed with respect to x over the interval [a, b].
02

Explain the Disk Method

The disk method is a technique used to find the volume of a solid of revolution generated by rotating a curve (the region under a curve) about an axis. In this method, the solid is divided into infinitesimally thin disks that are perpendicular to the axis of rotation. Each disk has a radius that is determined by the function, and the volume of each disk can be represented as: $$ dV = 蟺 [f(x)]^2 dx $$ where dV is the volume of an individual disk, f(x) is the function that defines the curve, and x represents the position along the axis of rotation. The volume of the entire solid can be found by integrating the volume of the individual disks: $$ V = \int_a^b 蟺 [f(x)]^2 dx $$
03

Comparing the Disk Method to General Slicing Method

The disk method has a specific approach to calculate the volume of a solid of revolution formed by rotating a curve around an axis, while the general slicing method calculates the volume of any solid using its cross-sectional areas. Observing the expressions for the volume of a solid given by both methods, we see similarities between the methods. In the disk method, the area of a single slice (a disk) is given by: $$ A(x) = 蟺 [f(x)]^2 $$ Comparing this to the general slicing method, it becomes clear that the disk method is just a special case of the general slicing method where the slices are disks and their area can be represented by the formula of the area of a circle. In other words, the disk method is the general slicing method applied specifically to solids of revolution formed by rotating curves around an axis. The disk method thus can be considered as a special case of the general slicing method in which the cross-sectional area function A(x) takes the form of the area of a circular disk with radius f(x).

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Most popular questions from this chapter

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