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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
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.5.9
Capitolo 6, Problema 6.5.9

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.)

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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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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.
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Arc Length of Parametric Curves

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.
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Derivatives of Other Trig Functions

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.
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Definition of the Definite Integral
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9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)


The work required to empty the top half of the tank

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


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9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


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For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.


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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 line.


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