9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

x = x³ ,y = 1, and x = 0; about the x-axis

검증된 단계별 안내

First, identify the region R bounded by the curves: the curve given by \(y = x^3\), the line \(y = 1\), and the vertical line \(x = 0\). Sketching this region helps visualize the problem.

Since the solid is generated by revolving the region about the x-axis, and the shell method is requested, we consider cylindrical shells formed by slicing the region parallel to the axis of revolution. Here, it is easier to use horizontal slices (in terms of \(y\)) because the shells will be vertical.

Express \(x\) as a function of \(y\) from the curve \(y = x^3\). Solving for \(x\) gives \(x = y^{1/3}\). This will help determine the radius and height of each shell.

Set up the shell method integral. The radius of a shell is the distance from the shell to the axis of rotation (the x-axis), which is \(y\). The height of the shell is the horizontal length of the region at height \(y\), which is from \(x=0\) to \(x = y^{1/3}\), so the height is \(y^{1/3} - 0 = y^{1/3}\).

The volume integral using the shell method is then \(V = 2\pi \int_{y=0}^{y=1} (\text{radius})(\text{height}) \, dy = 2\pi \int_0^1 y \cdot y^{1/3} \, dy\). Simplify the integrand and prepare to integrate over \(y\) from 0 to 1.

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주요 개념

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Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. Each shell's volume is found by multiplying its circumference, height, and thickness. This method is especially useful when the axis of rotation is parallel to the axis of the variable of integration.

Setting up the Region and Bounds

Identifying the region bounded by the given curves is crucial. Here, the region is bounded by y = x³, y = 1, and x = 0. Understanding these boundaries helps determine the limits of integration and the dimensions of each shell when revolving around the x-axis.

Revolution about the x-axis

Revolving a region about the x-axis means the shells are formed vertically with respect to y. The radius of each shell corresponds to the distance from the x-axis, and the height is determined by the horizontal length of the region at that y-value. This affects how the integral is set up.

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. x = x³ ,y = 1, and x = 0; about the x-axis