80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.

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Identify the region to be revolved: it is bounded by the curve \(y = \text{sech}(x)\), the x-axis (\(y=0\)), and the vertical lines \(x = -\ln\sqrt{3}\) and \(x = \ln\sqrt{3}\).

Recall the formula for the volume of a solid of revolution about the x-axis using the disk method: \(V = \pi \int_{a}^{b} [f(x)]^2 \, dx\), where \(f(x)\) is the function being revolved.

Set up the integral for the volume: \(V = \pi \int_{-\ln\sqrt{3}}^{\ln\sqrt{3}} \left( \text{sech}(x) \right)^2 \, dx\).

Use the identity \(\text{sech}(x) = \frac{1}{\cosh(x)}\), so the integrand becomes \(\text{sech}^2(x) = \frac{1}{\cosh^2(x)}\).

Evaluate the integral \(\int \text{sech}^2(x) \, dx\), which is a standard integral with antiderivative \(\tanh(x)\), and then apply the limits \(x = -\ln\sqrt{3}\) to \(x = \ln\sqrt{3}\) to find the volume.

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주요 개념

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

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The disk or washer method is commonly used, where the volume is computed by integrating the cross-sectional area perpendicular to the axis of rotation.

Hyperbolic Secant Function (sech(x))

The hyperbolic secant function, sech(x), is defined as 1/cosh(x), where cosh(x) = (e^x + e^{-x})/2. Understanding its properties and behavior is essential for setting up the integral limits and integrand when calculating the volume.

Definite Integration with Logarithmic Limits

The problem involves integrating between x = ±ln(√3), which requires understanding how to handle logarithmic limits in definite integrals. Proper evaluation of the integral at these bounds is crucial for obtaining the exact volume.