Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.4.1

Chapter 6, Problem 6.4.1

Assume f and g are continuous, with f(x) ≥ g(x) ≥ 0 on [a, b]. The region bounded by the graphs of f and g and the lines x=a and x=b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.

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Identify the region bounded by the curves y = f(x) and y = g(x), and the vertical lines x = a and x = b, where f(x) ≥ g(x) ≥ 0 on the interval [a, b].

Since the solid is formed by revolving this region about the y-axis, use the shell method, which involves cylindrical shells with radius equal to the distance from the y-axis, height equal to the difference between the functions, and thickness dx.

Express the radius of a typical shell as the x-value itself, since the distance from the y-axis to the shell at position x is simply x.

Express the height of the shell as the vertical distance between the two curves, which is f(x) - g(x).

Set up the integral for the volume using the shell method formula: \(V = \int_{a}^{b} 2\pi \times (\text{radius}) \times (\text{height}) \, dx = \int_{a}^{b} 2\pi x (f(x) - g(x)) \, dx\).

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Key Concepts

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

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. When revolving around the y-axis, each shell's radius is the x-value, height is the difference between the functions, and thickness is dx. The volume is found by integrating 2π(radius)(height) dx over [a, b].

Continuous Functions and Inequalities

Continuity of f and g on [a, b] ensures no gaps or jumps, allowing proper integration. The inequality f(x) ≥ g(x) ≥ 0 guarantees the region between the curves is well-defined and non-negative, which is essential for correctly determining the height of each shell.

Setting up the Integral Limits and Integrand

The integral limits correspond to the interval [a, b] on the x-axis. The integrand combines the shell radius (x) and height (f(x) - g(x)) multiplied by 2π. Correctly identifying these components is crucial to formulating the integral that represents the volume.

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Related Practice

Textbook Question

Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y=f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.

Textbook Question

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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Textbook Question

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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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=1−x^3, the x-axis, and the y-axis.

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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.

x=2−secy,x=2,y=π/3, and y=0; about x=2

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Textbook Question

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

x =2