Textbook Question
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
The region bounded by y = x²,y = 2x²−4x, and y = 0
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.R.66b
Surface area and volume Let f(x) = 1/3 x³ and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2].
b. Find the volume of the solid generated when R is revolved about the y-axis.
Verified step by step guidance1
Identify the region R bounded by the curve \(f(x) = \frac{1}{3}x^{3}\) and the x-axis on the interval \([0, 2]\). This means the region lies between \(x=0\) and \(x=2\), above the x-axis, and under the curve \(y = \frac{1}{3}x^{3}\).
Since the solid is generated by revolving the region R about the y-axis, use the method of cylindrical shells. The formula for the volume using cylindrical shells is:
\(V = \int_{a}^{b} 2\pi x \cdot f(x) \, dx\)
where \(x\) is the radius of the shell and \(f(x)\) is the height.
Substitute the given function \(f(x) = \frac{1}{3}x^{3}\) and the interval limits \(a=0\) and \(b=2\) into the volume integral:
\(V = \int_{0}^{2} 2\pi x \cdot \frac{1}{3}x^{3} \, dx\).
Simplify the integrand inside the integral:
\(2\pi x \cdot \frac{1}{3}x^{3} = \frac{2\pi}{3} x^{4}\), so the integral becomes
\(V = \int_{0}^{2} \frac{2\pi}{3} x^{4} \, dx\).
Evaluate the integral by integrating \(x^{4}\) with respect to \(x\), then multiply by the constant \(\frac{2\pi}{3}\), and finally apply the limits from 0 to 2 to find the volume.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method and the shell method. For rotation about the y-axis, the shell method is often more straightforward when the function is given in terms of x.
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Finding Volume Using Disks
Shell Method
The shell method calculates volume by summing cylindrical shells formed by revolving vertical slices of the region around the y-axis. Each shell's volume is 2π(radius)(height)(thickness), where radius is the distance from the y-axis, height is the function value, and thickness is a small change in x.
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Setting up Definite Integrals
To find the volume, you must express the integral with correct limits and integrand based on the region and axis of rotation. For the shell method, integrate with respect to x over [0, 2], using the radius and height functions derived from the problem's given function and interval.
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Related Practice
Textbook Question
Comparing volumes Let R be the region bounded by y=1/x^p and the x-axis on the interval [1, a], where p>0 and a>1 (see figure). Let Vₓ and Vᵧ be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively.
d. Find a general expression for Vᵧ in terms of a and p. Note that p=2 is a special case. What is Vᵧ when p=2?
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Textbook Question
Position, displacement, and distance A projectile is launched vertically from the ground at t=0, and its velocity in flight (in m/s) is given by v(t)=20−10t. Find the position, displacement, and distance traveled after t seconds, for 0≤t≤4.
Textbook Question
82–84. Fluid Forces Suppose the following plates are placed on a vertical wall so that the top of the plate is 2 m below the surface of a pool that is filled with water. Compute the force on each plate.
A circular plate with a radius of 2 m
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Textbook Question
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
Textbook Question
An area function Consider the functions y = x²/a and y = √x/a, where a>0. Find A(a), the area of the region between the curves.