9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)

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Rewrite the given curve equation in terms of \( y \): \( x = 2y - 4 \). Since \( x \) is expressed as a function of \( y \), we will use the arc length formula with respect to \( y \).

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

Calculate the derivative \( \frac{dx}{dy} \) from the given function: \[ \frac{dx}{dy} = \frac{d}{dy}(2y - 4) = 2 \]

Substitute \( \frac{dx}{dy} = 2 \) into the arc length integral and set the limits from \( y = -3 \) to \( y = 4 \): \[ L = \int_{-3}^{4} \sqrt{1 + (2)^2} \, dy = \int_{-3}^{4} \sqrt{1 + 4} \, dy = \int_{-3}^{4} \sqrt{5} \, dy \]

Evaluate the integral by factoring out the constant \( \sqrt{5} \): \[ L = \sqrt{5} \int_{-3}^{4} dy = \sqrt{5} [y]_{-3}^{4} = \sqrt{5} (4 - (-3)) = \sqrt{5} \times 7 \] This gives the arc length in terms of \( \sqrt{5} \) and the interval length.

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주요 개념

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Arc Length Formula

The arc length of a curve defined by a function can be found using the integral formula: L = ∫√(1 + (dy/dx)²) dx or L = ∫√(1 + (dx/dy)²) dy, depending on the variable of integration. This formula sums infinitesimal line segments along the curve to find its total length.

Parametric and Implicit Differentiation

When the curve is given in terms of y or implicitly, it is important to correctly find the derivative dx/dy or dy/dx. This may involve rearranging the equation or using implicit differentiation to express the derivative needed for the arc length formula.

Geometric Interpretation and Verification

After calculating the arc length using calculus, it is useful to verify the result geometrically if possible. For linear or simple curves, the arc length corresponds to the distance between endpoints, providing a way to check the correctness of the integral calculation.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)