Find the lengths of the curves in Exercises 19–22.
y = x¹/² ― (1/3) x³/² , 1 ≤ x ≤ 4

검증된 단계별 안내

Identify the function given: \(y = x^{\frac{1}{2}} - \frac{1}{3} x^{\frac{3}{2}}\) and the interval for \(x\) is \(1 \leq x \leq 4\).

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

Find the derivative \(\frac{dy}{dx}\) of the function: \[\frac{dy}{dx} = \frac{d}{dx} \left(x^{\frac{1}{2}} - \frac{1}{3} x^{\frac{3}{2}}\right)\] Use the power rule to differentiate each term.

Square the derivative to get \(\left(\frac{dy}{dx}\right)^2\) and then add 1 inside the square root: \[1 + \left(\frac{dy}{dx}\right)^2\]

Set up the integral for the arc length: \[L = \int_{1}^{4} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\] This integral can then be evaluated (analytically or numerically) to find the length of the curve.

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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 the Function

To apply the arc length formula, you must first find the derivative dy/dx of the given function y = x^(1/2) - (1/3)x^(3/2). This involves using power rule differentiation to handle fractional exponents accurately.

Evaluating Definite Integrals

After substituting dy/dx into the arc length integral, you evaluate the definite integral from x = 1 to x = 4. This may require algebraic simplification or numerical methods if the integral is complex or does not have a simple antiderivative.

Find the lengths of the curves in Exercises 19–22. y = x¹/² ― (1/3) x³/² , 1 ≤ x ≤ 4