9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = 3 ln x− x²/24 on [1, 6]

검증된 단계별 안내

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

Find the derivative \(\frac{dy}{dx}\) of the given function \(y = 3 \ln x - \frac{x^2}{24}\). Use the derivative rules for logarithmic and polynomial functions: \[\frac{dy}{dx} = 3 \cdot \frac{1}{x} - \frac{2x}{24}\]

Simplify the derivative expression: \[\frac{dy}{dx} = \frac{3}{x} - \frac{x}{12}\]

Substitute \(\frac{dy}{dx}\) into the arc length formula under the square root: \[L = \int_1^6 \sqrt{1 + \left(\frac{3}{x} - \frac{x}{12}\right)^2} \, dx\]

Set up the integral for evaluation. At this point, you can either simplify the expression inside the square root further or use numerical methods or a calculator to approximate the integral value over the interval \([1, 6]\).

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주요 개념

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

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is found using the integral formula L = ∫_a^b √(1 + (dy/dx)²) dx. This formula sums the lengths of infinitesimal line segments along the curve, providing the total distance traveled along it.

Derivative of the Function

To apply the arc length formula, you need the derivative dy/dx of the function y = 3 ln x − x²/24. The derivative measures the slope of the curve at each point, which is essential for calculating the integrand √(1 + (dy/dx)²).

Definite Integration

After finding the integrand, you evaluate the definite integral from x = 1 to x = 6. This process sums the continuous values of √(1 + (dy/dx)²) over the interval, yielding the exact arc length of the curve between the specified bounds.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.y = 3 ln x− x²/24 on [1, 6]