9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = −8x−3 on [−2, 6] (Use calculus.)

검증된 단계별 안내

Identify the function given: \(y = -8x - 3\) and the interval \([-2, 6]\) over which we want to find the arc length.

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

Compute the derivative \(\frac{dy}{dx}\) of the function \(y = -8x - 3\). Since this is a linear function, find \(\frac{dy}{dx}\) explicitly.

Substitute \(\frac{dy}{dx}\) into the arc length formula to get the integrand \(\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\), which will be a constant expression because the derivative is constant.

Evaluate the definite integral \(\int_{-2}^{6} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\) by integrating the constant integrand over the interval length \((6 - (-2))\).

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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 = f(x). The derivative represents the slope of the curve at any point, and its square is used inside the square root to account for the curve's steepness in the length calculation.

Definite Integration

Calculating arc length requires evaluating a definite integral over the given interval [a, b]. This process involves integrating the expression √(1 + (dy/dx)^2) with respect to x, which may require algebraic simplification or substitution to solve.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.y = −8x−3 on [−2, 6] (Use calculus.)