137. Find a curve through the origin in the xy-plane whose length from x = 0 to x = 1 is L = ∫ from 0 to 1 of sqrt(1 + (1/4)e^x) dx.

검증된 단계별 안내

Recognize that the problem asks for a curve \( y = f(x) \) passing through the origin \( (0,0) \) whose arc length from \( x=0 \) to \( x=1 \) is given by \( L = \int_0^1 \sqrt{1 + \frac{1}{4} e^x} \, dx \).

Recall the formula for the arc length of a curve \( y = f(x) \) from \( x=a \) to \( x=b \): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]

Compare the given integral's integrand \( \sqrt{1 + \frac{1}{4} e^x} \) with the arc length formula's integrand \( \sqrt{1 + (y')^2} \). This implies: \[ 1 + \left(\frac{dy}{dx}\right)^2 = 1 + \frac{1}{4} e^x \] which simplifies to: \[ \left(\frac{dy}{dx}\right)^2 = \frac{1}{4} e^x \]

Take the square root of both sides to find \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \pm \frac{1}{2} e^{x/2} \] Choose the positive root to find an increasing function (or negative if decreasing is desired).

Integrate \( \frac{dy}{dx} = \frac{1}{2} e^{x/2} \) with respect to \( x \) to find \( y(x) \): \[ y = \int \frac{1}{2} e^{x/2} \, dx + C \] Use the initial condition \( y(0) = 0 \) to solve for \( C \).

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주요 개념

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

Arc Length of a Curve

The arc length of a curve y = f(x) from x = a to x = b is given by the integral L = ∫ from a to b of sqrt(1 + (dy/dx)^2) dx. This formula measures the distance along the curve by summing infinitesimal line segments, accounting for both horizontal and vertical changes.

Differential Equation from Arc Length

Given an arc length integral involving sqrt(1 + (dy/dx)^2), we can form a differential equation by equating the integrand to sqrt(1 + (dy/dx)^2). Solving this equation for dy/dx allows us to find the function y(x) that produces the specified arc length.

Initial Condition and Integration

Since the curve passes through the origin, the initial condition y(0) = 0 is used to determine the constant of integration after solving the differential equation. Integrating dy/dx with this condition yields the explicit form of the curve y(x).