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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 6.6.5a
Chapter 6, Problem 6.6.5a

A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.


a. Using calculus

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Step 1: Recall the formula for the surface area of a solid of revolution about the x-axis. The formula is: A = 2π ∫[a,b] f(x)√(1 + (f'(x))²) dx, where f(x) is the function being revolved, and f'(x) is its derivative.
Step 2: Identify the function f(x) = 2 - x and its domain [0, 2]. Compute the derivative of f(x): f'(x) = -1.
Step 3: Substitute f(x) and f'(x) into the formula. The integrand becomes: 2π ∫[0,2] (2 - x)√(1 + (-1)²) dx. Simplify the square root term: √(1 + 1) = √2, so the integrand is 2π√2 ∫[0,2] (2 - x) dx.
Step 4: Break down the integral ∫[0,2] (2 - x) dx into two simpler integrals: ∫[0,2] 2 dx - ∫[0,2] x dx. Compute each integral separately using the power rule for integration.
Step 5: Multiply the result of the integral by 2π√2 to find the surface area. This completes the calculation process.

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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 a solid of revolution is calculated by revolving a curve around an axis. For a function f(x) revolved around the x-axis, the formula is given by A = 2π ∫[a to b] f(x) √(1 + (f'(x))^2) dx, where [a, b] is the interval of x. This formula accounts for both the height of the function and the slope, providing the total surface area generated.
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Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. It is denoted as ∫[a to b] f(x) dx and provides a numerical value that corresponds to the total area between the curve f(x) and the x-axis from x = a to x = b. Understanding how to evaluate definite integrals is crucial for calculating surface areas.
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Derivative and Its Role

The derivative of a function, denoted as f'(x), measures the rate of change of the function with respect to x. In the context of surface area, the derivative is used to find the slope of the curve, which is essential for the formula involving √(1 + (f'(x))^2). This term adjusts the length of the curve segment being revolved, ensuring accurate surface area calculations.
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Related Practice
Textbook Question

Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.


a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].

Textbook Question

Oil production An oil refinery produces oil at a variable rate given by Q'(t) = <1x3 matrix>, where is measured in days and is measured in barrels. 


a. How many barrels are produced in the first 35 days?

Textbook Question

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = 9−t² on [0, 4]; s(0)=−2

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Textbook Question

For the given regions R₁ and R₂, complete the following steps.


a. Find the area of region R₁.


R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.

Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

a. The displacement between t=0 and t=5

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.

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