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 axis.
y=√sin x,y=1, and x=0; about the x-axis
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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 6.6.7
Find the area of the surface generated when the given curve is revolved about the given axis.
y=3x+4, for 0≤x≤6; about the x-axis
Verified step by step guidance1
Step 1: Recall the formula for the surface area of a curve revolved about the x-axis: . Here, y is the function being revolved, and dy/dx is its derivative.
Step 2: Identify the function y = 3x + 4 and compute its derivative. The derivative of y with respect to x is .
Step 3: Substitute y = 3x + 4 and dy/dx = 3 into the formula. The integral becomes: .
Step 4: Simplify the square root term. Since , the integral simplifies to: .
Step 5: Evaluate the integral . Break it into two parts: and . Compute each part separately, then multiply the result by to find the surface area.
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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 revolution is calculated by revolving a curve around an axis. The formula involves integrating the circumference of infinitesimally small circular sections formed by the revolution. For a function y = f(x) revolved around the x-axis, the surface area S can be expressed as S = 2π ∫[a to b] f(x) √(1 + (f'(x))^2) dx, where f'(x) is the derivative of f(x).
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Example 1: Minimizing Surface Area
Definite Integral
A definite integral calculates the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. In the context of surface area, it helps determine the total surface area generated by the curve from x = 0 to x = 6. The result of a definite integral is a numerical value representing the total area or volume, depending on the application.
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05:43Definition of the Definite Integral
Parametric Representation of Curves
In some cases, curves can be represented parametrically, where both x and y are expressed as functions of a third variable, usually t. However, for the given linear function y = 3x + 4, a simple Cartesian representation suffices. Understanding how to manipulate and differentiate these representations is crucial for applying calculus concepts effectively in problems involving curves and their properties.
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Related Practice
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} y² = ln x,y² = ln x³, and y=2; about the x-axis
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Textbook Question
3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.
y = 2 cos 3x on [−π,π]
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Textbook Question
46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.
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Textbook Question
The region R is bounded by the graph of f(x)=2x(2−x) and the x-axis. Which is greater, the volume of the solid generated when R is revolved about the line y=2 or the volume of the solid generated when R is revolved about the line y=0? Use integration to justify your answer.
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Textbook Question
Find the volume of the torus formed when the circle of radius 2 centered at (3, 0) is revolved about the y-axis. Use geometry to evaluate the integral.
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