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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 6.3.47
Chapter 6, Problem 6.3.47

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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Step 1: Understand the problem. The region R is bounded by the curve y = 1 - x³, the x-axis, and the y-axis. We need to compare the volumes of solids generated when this region is revolved about the x-axis and the y-axis.
Step 2: Set up the volume formula for revolution about the x-axis. Use the disk method, where the volume is given by: V = ∫[a,b] π(y)^2 dx. Here, y = 1 - x³, and the limits of integration are from x = 0 to x = 1 (since the region is bounded by the y-axis and the curve).
Step 3: Set up the volume formula for revolution about the y-axis. Use the shell method, where the volume is given by: V = ∫[a,b] 2πx(y) dx. Again, y = 1 - x³, and the limits of integration are from x = 0 to x = 1.
Step 4: Compare the integrals. For the x-axis revolution, the integral involves squaring the function y = 1 - x³, resulting in π∫[0,1] (1 - x³)^2 dx. For the y-axis revolution, the integral involves multiplying x by the function y = 1 - x³, resulting in 2π∫[0,1] x(1 - x³) dx. Both integrals need to be evaluated to determine which volume is greater.
Step 5: Analyze the results conceptually. The volume about the x-axis depends on the square of the height of the region (y-values), while the volume about the y-axis depends on the product of the radius (x-values) and the height (y-values). This comparison requires evaluating the integrals, but the setup allows us to proceed systematically.

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Key Concepts

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

Volume of Revolution

The volume of revolution refers to the volume of a solid formed by rotating a two-dimensional area around an axis. This can be calculated using methods such as the disk method or the washer method, depending on whether the solid has a hole in it. The choice of axis (x-axis or y-axis) significantly affects the volume calculation, as it determines the shape and dimensions of the resulting solid.
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Finding Volume Using Disks

Disk and Washer Methods

The disk and washer methods are techniques used to calculate the volume of solids of revolution. The disk method is applied when the solid has no hole, using the formula V = π∫[f(x)]²dx for rotation about the x-axis. The washer method is used when there is a hole, incorporating an outer and inner radius, and is expressed as V = π∫([R(x)]² - [r(x)]²)dx, allowing for the calculation of volumes for more complex shapes.
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Disk Method Using y-Axis

Bounded Regions

Bounded regions in calculus refer to areas enclosed by curves and lines, which can be analyzed for various properties, including area and volume. In this question, the region R is defined by the curve y = 1 - x³, the x-axis, and the y-axis. Understanding the boundaries of R is crucial for setting up the correct integrals to compute the volumes when the region is revolved around the x-axis or y-axis.
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Area of Polar Regions
Related Practice
Textbook Question

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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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

Leaky Bucket A 1-kg bucket resting on the ground contains 3 kg of water. How much work is required to raise the bucket vertically a distance of 10 m if water leaks out of the bucket at a constant rate of 1/5 kg/m? Assume the weight of the rope used to raise the bucket is negligible. (Hint: Use the definition of work, W = ∫a^bF(y) dy, where F is the variable force required to lift an object along a vertical line from y=a to y=b.)

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Textbook Question

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


The region bounded by x=y(y−1) and y=x/3

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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. 


{Use of Tech} y = √sin^−1x,y = √π/2, and x=0; about the x-axis

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