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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.5.17
Chapter 6, Problem 6.5.17

9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = y⁴/4 + 1/8y², for 1≤y≤2

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Step 1: Recall the formula for arc length of a curve given in parametric form: L = ∫ab √(1 + (dx/dy)2) dy. Here, the curve is given as x = y4/4 + 1/(8y2), and the interval is 1 ≤ y ≤ 2.
Step 2: Compute the derivative dx/dy. Differentiate x = y4/4 + 1/(8y2) with respect to y. Use the power rule and chain rule: dx/dy = y3 - 1/(4y3).
Step 3: Square the derivative dx/dy. This gives (dx/dy)2 = (y3 - 1/(4y3))2. Expand the square if necessary for simplification.
Step 4: Substitute (dx/dy)2 into the arc length formula. The integrand becomes √(1 + (y3 - 1/(4y3))2). Set up the integral: L = ∫12 √(1 + (y3 - 1/(4y3))2) dy.
Step 5: Evaluate the integral. Depending on the complexity of the integrand, you may need numerical methods or advanced techniques to compute the arc length. Ensure proper limits of integration are applied from y = 1 to y = 2.

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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 defined by a function can be calculated using the formula L = ∫√(1 + (dy/dx)²) dy, where dy/dx is the derivative of the function with respect to y. This formula allows us to find the length of the curve over a specified interval by integrating the square root of one plus the square of the derivative.
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Arc Length of Parametric Curves

Implicit Differentiation

When dealing with curves defined in terms of y, implicit differentiation is often used to find dy/dx. This technique involves differentiating both sides of an equation with respect to y, treating x as a function of y, which is essential for applying the arc length formula correctly.
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Finding The Implicit Derivative

Definite Integrals

Definite integrals are used to calculate the total length of the curve over a specific interval. In this context, the limits of integration correspond to the values of y (from 1 to 2) for which the arc length is being calculated, allowing us to evaluate the integral to find the exact length of the curve.
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Definition of the Definite Integral
Related Practice
Textbook Question

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=2 sin x and y=0 on [0,π]; about y=−2

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Textbook Question

Find the area of the surface generated when the given curve is revolved about the given axis.


y=x^3/2−x^1/2 / 3, for 1≤x≤2; about the x-axis 

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Textbook Question

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.


R is bounded by y=4−2x, the x-axis, and the y-axis.

Textbook Question

Determine the area of the shaded region in the following figures.

Textbook Question

Find the area of the surface generated when the given curve is revolved about the given axis.


y=4x−1, for 1≤x≤4; about the y-axis (Hint: Integrate with respect to y.)

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Textbook Question

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.

y = (1+x²)^−1,y = 0,x = 0, and x = 2; about the y-axis