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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 6.6.7
Chapter 6, Problem 6.6.7

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


y=3x+4, for 0≤x≤6; about the x-axis

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Step 1: Recall the formula for the surface area of a curve revolved about the x-axis: A = 2π ∫[a,b] y √(1 + (dy/dx)2) dx. Here, y is the function being revolved, and dy/dx is its derivative.
Step 2: Identify the function y = 3x + 4 and compute its derivative. The derivative of y with respect to x is dy/dx = 3.
Step 3: Substitute y = 3x + 4 and dy/dx = 3 into the formula. The integral becomes: A = 2π ∫[0,6] (3x + 4) √(1 + 32) dx.
Step 4: Simplify the square root term. Since √(1 + 32) = √10, the integral simplifies to: A = 2π √10 ∫[0,6] (3x + 4) dx.
Step 5: Evaluate the integral ∫[0,6] (3x + 4) dx. Break it into two parts: ∫[0,6] 3x dx and ∫[0,6] 4 dx. Compute each part separately, then multiply the result by 2π √10 to find the surface area.

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Key Concepts

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

Surface Area of Revolution

The surface area of revolution is calculated by revolving a curve around an axis. The formula involves integrating the circumference of infinitesimally small circular sections formed by the revolution. For a function y = f(x) revolved around the x-axis, the surface area S can be expressed as S = 2π ∫[a to b] f(x) √(1 + (f'(x))^2) dx, where f'(x) is the derivative of f(x).
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Example 1: Minimizing Surface Area

Definite Integral

A definite integral calculates the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. In the context of surface area, it helps determine the total surface area generated by the curve from x = 0 to x = 6. The result of a definite integral is a numerical value representing the total area or volume, depending on the application.
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Definition of the Definite Integral

Parametric Representation of Curves

In some cases, curves can be represented parametrically, where both x and y are expressed as functions of a third variable, usually t. However, for the given linear function y = 3x + 4, a simple Cartesian representation suffices. Understanding how to manipulate and differentiate these representations is crucial for applying calculus concepts effectively in problems involving curves and their properties.
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Related Practice
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. 


{Use of Tech} y² = ln x,y² = ln x³, and y=2; about the x-axis

Textbook Question

60–63. Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ________.


v(t) = t(25−t²)^1/2, for 0≤t≤5

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Textbook Question

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

x=4y^3/2−y^1/2 / 12, for 1≤y≤4; about the y-axis

Textbook Question

3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.

y = 2 cos 3x on [−π,π]

Textbook Question

The region R is bounded by the graph of f(x)=2x(2−x) and the x-axis. Which is greater, the volume of the solid generated when R is revolved about the line y=2 or the volume of the solid generated when R is revolved about the line y=0? Use integration to justify your answer.

Textbook Question

Find the volume of the torus formed when the circle of radius 2 centered at (3, 0) is revolved about the y-axis. Use geometry to evaluate the integral.

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