Assume f and g are continuous, with f(x) ≥ g(x) ≥ 0 on [a, b]. The region bounded by the graphs of f and g and the lines x=a and x=b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.

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Identify the region bounded by the curves y = f(x) and y = g(x), and the vertical lines x = a and x = b, where f(x) ≥ g(x) ≥ 0 on the interval [a, b].

Since the solid is formed by revolving this region about the y-axis, use the shell method, which involves cylindrical shells with radius equal to the distance from the y-axis, height equal to the difference between the functions, and thickness dx.

Express the radius of a typical shell as the x-value itself, since the distance from the y-axis to the shell at position x is simply x.

Express the height of the shell as the vertical distance between the two curves, which is f(x) - g(x).

Set up the integral for the volume using the shell method formula: \(V = \int_{a}^{b} 2\pi \times (\text{radius}) \times (\text{height}) \, dx = \int_{a}^{b} 2\pi x (f(x) - g(x)) \, dx\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. When revolving around the y-axis, each shell's radius is the x-value, height is the difference between the functions, and thickness is dx. The volume is found by integrating 2π(radius)(height) dx over [a, b].

Continuous Functions and Inequalities

Continuity of f and g on [a, b] ensures no gaps or jumps, allowing proper integration. The inequality f(x) ≥ g(x) ≥ 0 guarantees the region between the curves is well-defined and non-negative, which is essential for correctly determining the height of each shell.

Setting up the Integral Limits and Integrand

The integral limits correspond to the interval [a, b] on the x-axis. The integrand combines the shell radius (x) and height (f(x) - g(x)) multiplied by 2π. Correctly identifying these components is crucial to formulating the integral that represents the volume.