Textbook Question
Force on the end of a tank Determine the force on a circular end of the tank in Figure 6.78 if the tank is full of gasoline. The density of gasoline is ρ = 737 kg/m³.
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 6.3.28
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=x,y=2x, and y=6 ; about the y-axis
Verified step by step guidance1
Identify the region R bounded by the curves y = x, y = 2x, and y = 6. This region lies between the two lines y = x and y = 2x, and is capped by the horizontal line y = 6.
Since the solid is generated by revolving the region R about the y-axis, express the boundaries in terms of x as functions of y. From y = x, we get x = y, and from y = 2x, we get x = y/2.
Determine the limits of integration for y, which are from y = 0 (where the lines intersect at the origin) up to y = 6 (the horizontal boundary).
Use the method of cylindrical shells to find the volume. The formula for the volume when revolving around the y-axis is:
\[ V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dy \]
Here, the radius is the distance from the y-axis to the shell, which is x, and the height is the difference between the two x-values (outer radius minus inner radius) at each y.
Set up the integral for the volume as:
\[ V = 2\pi \int_{0}^{6} y \left( \frac{y}{2} - y \right) dy \]
This integral represents the volume of the solid formed by revolving the region R about the y-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Setting up the region bounded by curves
Understanding the region R requires identifying the intersection points and boundaries formed by the curves y = x, y = 2x, and y = 6. This involves solving for x in terms of y or vice versa to describe the limits of integration accurately.
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Volume of solids of revolution about the y-axis
When a region is revolved around the y-axis, the volume can be found using methods like the shell method or the washer method. The shell method integrates cylindrical shells parallel to the axis, while the washer method integrates cross-sectional disks perpendicular to the axis.
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06:30Disk Method Using y-Axis
Using the shell method for volume calculation
The shell method is often convenient when revolving around the y-axis, especially when the region is described in terms of y. It involves integrating 2π(radius)(height) with respect to x, where radius is the distance from the y-axis and height is the vertical length of the shell.
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04:48Finding Volume Using Disks
Related Practice
Textbook Question
53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x,y = 2x+2,x = 2, and x=6; about the y-axis
Textbook Question
A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where 0≤h≤8 0 ≤ ℎ ≤ 8 . Find the volume of water in the bowl as a function of h. (Check the special cases h=0 and h=8.)
Textbook Question
Use the general slicing method to find the volume of the following solids.
The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares
Textbook Question
Function defined as an integral Write the integral that gives the length of the curve y = f(x) = ∫₀^x sin t dt on the interval [0,π]
Textbook Question
13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.
ρ(x) = 5e^-2x,for 0≤x≤4