58–61. Arc length Find the length of the following curves.
y = 2x+4 on [−2,2] (Use calculus.)

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

Identify the function and the interval: here, \(y = 2x + 4\) and the interval is \([-2, 2]\).

Compute the derivative of \(y\) with respect to \(x\): \[\frac{dy}{dx} = \frac{d}{dx}(2x + 4)\]

Substitute the derivative into the arc length formula: \[L = \int_{-2}^{2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]

Evaluate the integral to find the length of the curve over the given interval.

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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 calculates the length by summing infinitesimal line segments along the curve.

Derivative of a Function

The derivative dy/dx represents the slope of the curve at any point x. For the arc length formula, the derivative is squared and added to 1 inside the square root to account for both horizontal and vertical changes along the curve.

Definite Integration

Definite integration evaluates the integral over a specific interval [a, b], providing a numerical value for the arc length. It sums the continuous contributions of small segments between the given bounds.

58–61. Arc length Find the length of the following curves.y = 2x+4 on [−2,2] (Use calculus.)