9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = x^3/2 / 3 − x^1/2 on [4, 16]

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Identify the function given: \(y = \frac{x^{3/2}}{3} - x^{1/2}\), and the interval \([4, 16]\) over which we want to find the arc length.

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

Find the derivative \(\frac{dy}{dx}\) of the function: First, rewrite the function as \(y = \frac{1}{3} x^{3/2} - x^{1/2}\). Then differentiate term-by-term using the power rule.

Square the derivative \(\left(\frac{dy}{dx}\right)^2\) and add 1 inside the square root to form the integrand: \(\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\)

Set up the definite integral for the arc length over the interval \([4, 16]\): \(L = \int_4^{16} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\). Evaluate this integral to find the arc length.

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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 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 must compute the derivative dy/dx of the given function y = (x^(3/2))/3 − x^(1/2). This involves using power rule differentiation for fractional exponents to find the slope of the curve at any point.

Definite Integration

After finding the integrand √(1 + (dy/dx)^2), evaluate the definite integral from x = 4 to x = 16. This requires integrating the expression over the interval to obtain the exact arc length, often involving substitution or numerical methods if the integral is complex.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.y = x^3/2 / 3 − x^1/2 on [4, 16]