Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problema 6.5.8

Capitolo 6, Problema 6.5.8

Find the arc length of the line y = 4−3x on [−3, 2] 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, \(y = 4 - 3x\) and the interval is \([-3, 2]\).

Compute the derivative \(\frac{dy}{dx}\): since \(y = 4 - 3x\), then \[\frac{dy}{dx} = -3\]

Substitute the derivative into the arc length formula: \[L = \int_{-3}^{2} \sqrt{1 + (-3)^2} \, dx = \int_{-3}^{2} \sqrt{1 + 9} \, dx = \int_{-3}^{2} \sqrt{10} \, dx\]

Evaluate the integral: since \(\sqrt{10}\) is constant, the integral becomes \[L = \sqrt{10} \times (2 - (-3)) = \sqrt{10} \times 5\]. To verify using geometry, recognize that the graph is a straight line segment between points \((-3, 4 - 3(-3))\) and \((2, 4 - 3(2))\). Calculate the distance between these points using the distance formula \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] and confirm it matches the arc length.

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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.

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06:29

Arc Length of Parametric Curves

Derivative of a Function

The derivative dy/dx represents the slope of the function y = f(x) 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.

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Derivatives of Other Trig Functions

Geometric Interpretation of a Line Segment

Since y = 4 - 3x is a straight line, its arc length over [−3, 2] can be found using the distance formula between two points: √((x2 - x1)^2 + (y2 - y1)^2). This provides a geometric verification of the calculus result.

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Percorso guidato

05:13

Slopes of Tangent Lines

Pratica correlata

Domanda del libro di testo

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=x^2,y=2−x, and y=0; about the y-axis

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Domanda del libro di testo

45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines.

x = -1

110

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Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.

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Find the area of the region described in the following exercises.

The region bounded by y=4x+4, y=6x+6, and x=4

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9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2

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Domanda del libro di testo

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.

y=x and y=1+x/2; about y=3

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