Find the lengths of the curves in Exercises 19–22.
y = (5/12) x⁶/⁵ ― (5/8)x⁴/⁵ , 1 ≤ x ≤ 32

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

Identify the function given: \[y = \frac{5}{12} x^{\frac{6}{5}} - \frac{5}{8} x^{\frac{4}{5}}\] and the interval: \[1 \leq x \leq 32\]

Find the derivative \(\frac{dy}{dx}\) by differentiating each term using the power rule: \[\frac{dy}{dx} = \frac{5}{12} \cdot \frac{6}{5} x^{\frac{6}{5} - 1} - \frac{5}{8} \cdot \frac{4}{5} x^{\frac{4}{5} - 1}\] Simplify the coefficients and exponents carefully.

Substitute \(\frac{dy}{dx}\) into the arc length formula under the square root: \[\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\] This expression will be the integrand for the length integral.

Set up the definite integral for the length of the curve: \[L = \int_{1}^{32} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\] At this point, you can simplify the integrand if possible, then evaluate the integral using appropriate methods (analytical or numerical).

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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 L = ∫_a^b √(1 + (dy/dx)²) dx. This formula calculates the distance along the curve by summing infinitesimal line segments, accounting for both horizontal and vertical changes.

Derivative of a Function

The derivative dy/dx represents the rate of change or slope of the function y with respect to x. For the arc length, computing dy/dx is essential to determine how steep the curve is at each point, which affects the length calculation.

Integration Techniques for Power Functions

Since the function involves fractional exponents, integrating the arc length expression requires familiarity with integrating power functions and possibly simplifying the integrand. Understanding how to handle fractional powers and algebraic manipulation is key to solving the integral.

Find the lengths of the curves in Exercises 19–22. y = (5/12) x⁶/⁵ ― (5/8)x⁴/⁵ , 1 ≤ x ≤ 32