Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.

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Identify the curve and the interval: The curve is given by \(y = \cosh x\) on the interval \([-\ln 2, \ln 2]\).

Recall the formula for the surface area of a surface of revolution about the x-axis: \(S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\)

Compute the derivative of \(y = \cosh x\): \(\frac{dy}{dx} = \sinh x\)

Substitute \(y\) and \(\frac{dy}{dx}\) into the surface area formula: \(S = \int_{-\ln 2}^{\ln 2} 2\pi \cosh x \sqrt{1 + (\sinh x)^2} \, dx\)

Simplify the expression inside the square root using the identity \(1 + \sinh^2 x = \cosh^2 x\), so the integrand becomes \(2\pi \cosh x \cdot \cosh x = 2\pi \cosh^2 x\). Then set up the integral \(S = \int_{-\ln 2}^{\ln 2} 2\pi \cosh^2 x \, dx\) to be evaluated.

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주요 개념

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

Surface Area of a Surface of Revolution

The surface area generated by revolving a curve y = f(x) about the x-axis is found using the integral formula S = ∫ 2π y √(1 + (dy/dx)^2) dx over the given interval. This formula accounts for the circumference of circular slices and the curve's slope.

Hyperbolic Cosine Function and Its Derivative

The catenary is defined by y = a cosh(x/a), where cosh x = (e^x + e^{-x})/2. Its derivative, sinh x, is essential for computing the slope dy/dx, which appears in the surface area integral. Understanding these functions helps evaluate the integral accurately.

Evaluating Definite Integrals with Logarithmic Limits

The problem involves integrating over the interval [–ln 2, ln 2]. Recognizing properties of logarithms and symmetry can simplify the integral. Proper evaluation of definite integrals with such limits is crucial for obtaining the exact surface area.