Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.5.7

Chapter 6, Problem 6.5.7

Find the arc length of the line y = 2x+1 on [1, 5] using calculus and verify your answer using geometry.

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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 interval: here, \(y = 2x + 1\) and the interval is \([1, 5]\).

Compute the derivative \(\frac{dy}{dx}\): since \(y = 2x + 1\), then \[\frac{dy}{dx} = 2\]

Substitute \(\frac{dy}{dx}\) into the arc length formula: \[L = \int_1^5 \sqrt{1 + (2)^2} \, dx = \int_1^5 \sqrt{1 + 4} \, dx = \int_1^5 \sqrt{5} \, dx\] Since \(\sqrt{5}\) is constant, the integral simplifies to \[L = \sqrt{5} \int_1^5 dx\]

Evaluate the integral and interpret the result geometrically: The integral \(\int_1^5 dx\) equals \((5 - 1) = 4\), so \[L = 4 \sqrt{5}\] To verify geometrically, recognize that \(y = 2x + 1\) is a straight line segment between points \((1, 3)\) and \((5, 11)\). Use the distance formula \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Calculate this distance and confirm it matches the arc length found via calculus.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 of the curve by summing infinitesimal line segments along the curve.

Derivative of a Function

The derivative dy/dx represents the slope of the function y = f(x) at any point x. For the line y = 2x + 1, the derivative is constant, which simplifies the arc length calculation since the slope does not change over the interval.

Geometric Interpretation of Line Segment Length

The length of a straight line segment between two points (x1, y1) and (x2, y2) can be found using the distance formula √((x2 - x1)^2 + (y2 - y1)^2). This provides a geometric verification of the arc length calculated via calculus for linear functions.

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